ARITHMATIC PROGRESSIONS

ARITHMATIC PROGRESSIONS

6.1 PROGRESSIONS : Those sequence whose terms follow certain patterns are called progression. Generally there are three types of progression.

(i) Arithmetic Progression (A.P.)

(ii) Geometric Progression (G.P.)

(iii) Harmonic Progression (H.P.)

ARITHMATIC PROGRESSIONS

6.2 ARTHMETIC PROGRESSION :

A sequence is called an A.P., if the difference of a term and the previous term is always same. i.e. $\mathbf{d}={\mathbf{t}}_{\mathbf{n}+1}$ - ${\mathbf{t}}_n=$ Constant for all $n\in N$. The constant difference, generally denoted by ’ $d$ ’ is called the common difference.

ARITHMATIC PROGRESSIONS

Ex. 1 Find the common difference of the following A.P. : 1,4,7,10,13,16 ……

ARITHMATIC PROGRESSIONS

Sol. $4-1=7-4=10-7=13-10=16-13=3$ (constant). $\therefore$ Common difference $(d)=3$.

ARITHMATIC PROGRESSIONS

6.3 GENERAL FORM OF AN A.P.:

If we denote the starting number i.e. the $1^{\text{st }}$ number by ’ $\mathbf{a}$ ’ and a fixed number to the added is ’ $\mathbf{d}$ ’ then $\mathbf{a}$, $\mathbf{a}$ $+d,a+2d,a+3d,a+4d$ forms an A.P.

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Ex. 2 Find the A.P. whose $1^{\text{st }}$ term is 10 & common difference is 5.

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Sol. $$ Given : First term $(a)=10$ & Common difference $(d)=5.\therefore$ A.P. is $10,15,20,25,30,\dots .$.

ARITHMATIC PROGRESSIONS

6.4 ${\mathbf{n}}^{\text{th }}$ TERM OF AN A.P.:

Let A.P. be $a,a+d,a+2d,a+3d,\dots .$.

Then, First term $({\mathbf{a}}_1)=a+0.d$

Second term $(a_2)=a+1.d$

Third term $({\mathbf{a}}_3)=a+2.d$

$n^{\text{th }}$ term $({\mathbf{a}}_{\mathbf{n}})=a+(n-1)d$

$\therefore {\mathbf{a}}_{\mathbf{n}}=\mathbf{a}+(\mathbf{n}-1)d$ is called the $n^{\text{th }}$ term.

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Ex. 3 Determine the A.P. whose their term is 16 and the difference of $5^{\text{th }}$ term from $7^{\text{th }}$ term is 12.

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Sol. Given: $a_3=a+(3-1)d=a+2d=16$

$a_7-a_5=12$

$(a+6d)-(a+4d)=12$

$a+6d-a-4d=12$

$2d=12$

$a=16-12$

Put $d=6$ in equation (i)

$a=4\therefore$ A.P. is $4,10,16,22,28,\dots \dots$.

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Ex. 4 Which term of the sequence 72, 70, 68, 66, ….. is 40 ?

ARITHMATIC PROGRESSIONS

Sol. Here $1^{\text{st }}$ term $x=72$ and common difference $d=70-72=-2$

$\therefore$ For finding the value of $n$

$a_n=a+(n-1)d$

$\Rightarrow 40=72+(n-1)(-2)$

$\Rightarrow 40-72=-2n+2$

$\Rightarrow -32=-2n+2$

$\Rightarrow -34=-2n$

$\Rightarrow n=17$

$\therefore {17}^{\text{th }}$ term is 40 .

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Ex. 5 Is 184, a term of the sequence 3,7,11,…. ?

ARITHMATIC PROGRESSIONS

Sol. Here $1^{\text{st }}$ term $(a)=3$ and common difference $(d)=7-3=4$ $n^{\text{th }}$ term $(a_n)=a+(n-1)d$

$\Rightarrow 184=3+(n-1)4\Rightarrow 181=4n-4\Rightarrow 185=4n$

$\Rightarrow n=\frac{185}{4}$ Since, $n$ is not a natural number.

$\therefore 184$ is not a term of the given sequence.

ARITHMATIC PROGRESSIONS

Ex. 6 Which term of the sequence 20, $19\frac{1}{2},18\frac{1}{2},17\frac{3}{4}$ is the 1 st negative term.

ARITHMATIC PROGRESSIONS

Sol. Here $1^{\text{st }}$ term $(a)=20$, common difference $(d)=19\frac{1}{4}-20=-\frac{3}{4}$

Let $n^{\text{th }}$ term of the given A.P. be $1^{\text{st }}$ negative term $\therefore a_n<0$

i.e. $a+(n-1)d<0$

$\Rightarrow 20+(n-1)(-\frac{3}{4})<0\Rightarrow \frac{83}{4}-\frac{3n}{4}<0\Rightarrow 3n>83\Rightarrow n>\frac{83}{3}\Rightarrow n>27\frac{2}{3}$

Since, 28 is the natural number just greater then $27\frac{2}{3}$.

$\therefore 1^{\text{st }}$ negative term is ${28}^{\text{th }}$.

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Ex. 7 If $p^{\text{th }},q^{\text{th }}$ and $r^{\text{th }}$ term of an A.P. are $a,b,c$ respectively, then show than $a(q-r)+b(-p)+c(p-q)=0$.

ARITHMATIC PROGRESSIONS

Sol. $a_p=a\Rightarrow A+(p-1)D=a$

$a_q=b\Rightarrow$

$A+(q-1)D=b$

$a_r=c\Rightarrow$

$A+(r+1)D=c$

Now, L.H.S.

$=a(q-r)+b(r-p)+c(p-q)$

$=A+(p-1)D(q-r)+A+(q-1)D(r-p)+A+(r-1)D(p-q)$

$=0$. R.H.S

ARITHMATIC PROGRESSIONS

Ex. 8 If $m$ times the $m^{\text{th }}$ term of an A.P. is equal to $n$ times its ${}^{th}$ term. Show that the $(m+n{)}^{th}$ term of the A.P.

ARITHMATIC PROGRESSIONS

Sol. Let $A$ the $1^{\text{st }}$ term and $D$ be the common difference of the given A.P.

Then, $m{a}_m=n{a}_n$

$\Rightarrow m[A+(m-1)D]=n[A+(n-1)D]$

$\begin{array}{cc}\Rightarrow & A(m-1)+D[m+n(m-n)-(m-n)]=0\\ \Rightarrow & a_{m+n}=0\end{array}$

$\Rightarrow A+(m+n-1)D=0$

ARITHMATIC PROGRESSIONS

Ex. 9 If the $p^{\text{th }}$ term of an A.P. is $q$ and the $q^{\text{th }}$ term is $p$, prove that $it{s}^{\text{th }}$ term is $(p+q-n)$.

ARITHMATIC PROGRESSIONS

Sol. $a_p=q\Rightarrow A+(p-1)D=q$

& $a_q=p\Rightarrow A+(q-1)D=p$

Solve (i) & (ii) to get $D=-1$ & $A=p+q-1$

$\therefore a_n=A+(n-1)D$

$a_n=(p+q-1)+(n-1)(-1)a_n=p+q-n$.

ARITHMATIC PROGRESSIONS

Ex. 10 If the $m^{\text{th }}$ term of an A.P. $\frac{1}{n}$ and $n^{\text{th }}$ term be $\frac{1}{m}$ then show that its $(mn)$ term is 1 .

ARITHMATIC PROGRESSIONS

Sol.

$a_m=\frac{1}{n}\Rightarrow A+(m-1)D=\frac{1}{n}\dots \dots \dots \dots (i)$

$a_m=\frac{1}{m}\Rightarrow A+(n-1)D=\frac{1}{m}\dots \dots \dots \dots (ii)$

By solving (i) & (ii) $D=\frac{1}{mn}$ & $A=\frac{1}{mn}$

$\therefore a_{mn}=A+(mn-1)D=1$.

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6.5 $m^{\text{th }}$ TERM OF AN A.P. FROM THE END :

Let ’ $\mathbf{a}$ ’ be the $1^{\text{st }}$ term and ’ $\mathbf{d}$ ’ be the common difference of an A.P. having $\mathbf{n}$ terms. Then ${\mathbf{m}}^{\text{th }}$ term from the end is $(\mathbf{n}-\mathbf{m}+\mathbf{1}{)}^{\text{th }}$ term from beginning or ${\mathbf{n}-(\mathbf{m}-)}^{\text{th }}$ term from beginning.

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Ex. 11 Find ${20}^{\text{th }}$ term from the end of an A.P. 3,7,11….. 407.

ARITHMATIC PROGRESSIONS

Sol. $407=3+(n-1)4\Rightarrow n=102$

$\therefore {20}^{\text{th }}$ term from end $\Rightarrow m=20$

$a_{102-(20-1)}=a_{102-19}=a_{83}$ from the beginning.

$a_{83}=3+(83+1)4=331$.

ARITHMATIC PROGRESSIONS

6.6 SELECTION OF TERMS IN AN A.P. :

Sometimes we require certain number of terms in A.P. The following ways of selecting terms are generally very convenient.

ARITHMATIC PROGRESSIONS

Ex. 12 The sum of three number in A.P. is -3 and their product is 8. Find the numbers.

ARITHMATIC PROGRESSIONS

Sol. Three no. ’s in A.P. be $a-d,a,a+d$

$\begin{array}{rl}& \therefore a-d+a+a+d=-3\\ & 3a=-3\Rightarrow a=-1\\ & (a-d)a(a+d)=8\\ & a(a^2-d^2)=8\end{array}$

$\begin{array}{rl}& (-1)(1-d^2)=8\\ & 1-d^2=-8\Rightarrow d^2=9\Rightarrow d=\pm 3\end{array}$

$\text{ If }a=8\text{and}d=3\text{ numbers are }-4,-1,2.\text{ If }a=8a=8\text{and}d=-\text{ numbers are }2,-1,-4\text{. }$

ARITHMATIC PROGRESSIONS

6.7 SUM OF n TERMS OF AN A.P. :

Let A.P. be $a,a+d,a+1d,a+3d,\dots ...a+(n-1)d$

Then, $S_n=a+(a+d)+\dots ..+a+(n-2)d+a+(n-1)d$

also, $S_n=a+(n-1)d+a+(n-2)d+\dots \dots +(a+d)+a$

Add (i) & (ii)

$\Rightarrow 2S_n=2a+(n-1)d+2a+(n-1)d+\dots \dots \dots \dots \dots +2a+(n-1)d$

$\Rightarrow 2S_n=n[2a+(n-1)d]\Rightarrow S_n=\frac{n}{2}[2a+(n+1)d]$

$S_n=\frac{n}{2}[a+a+(n-1)d]=\frac{n}{2}[a+\ell ]\therefore S_n=\frac{n}{2}[a+\ell ]$ where $\ell$ is the last term.

ARITHMATIC PROGRESSIONS

Ex. 13 Find the sum of 20 terms of the A.P. 1,4,7,10…..

ARITHMATIC PROGRESSIONS

Sol. $a=1,d=3S_n=\frac{n}{2}[2a+(n-1)d]S_{20}=\frac{20}{2}[2(1)+(20-1)3]$

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Ex. 14 Find the sum of all three digit natural numbers. Which are divisible by 7.

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Sol. $1^{\text{st }}$ no. is 105 and last no. is 994 .

Find $n994=105+(n+1)7$

$\therefore n=128\therefore \text{ Sum, }{S}_{128}=\frac{128}{2}[105+994]$

ARITHMATIC PROGRESSIONS

6.8 PROPERTIES OF A.P. :

(A) For any real numbers $a$ and $b$, the sequence whose $n^{\text{th }}$ term is ${\mathbf{a}}_{\mathbf{n}}=\mathbf{a}\mathbf{+}\mathbf{b}$ is always an A.P. with common difference ’ $\mathbf{a}$ ’ (i.e. coefficient of term containing $\mathbf{n}$ )

(B) If any $n^{\text{th }}$ term of sequence is a linear expression in $\mathbf{n}$ then the given sequence is an A.P.

(C) If a constant term is added to or subtracted from each term of an A.P. then the resulting sequence is also an A.P. with the same common difference.

(D) If each term of a given A.P. is multiplied or divided by a non-zero constant $K$, then the resulting sequence is also an A.P. with common difference $\mathbf{K}\mathbf{d}$ or respectively. Where $\mathbf{d}$ is the common difference of the given A.P.

(E) In a finite A.P. the sum of the terms equidistant from the beginning and end is always same and is equal to the sum of $1^{\text{st }}$ and last term.

(F) If three numbers $\mathbf{a},\mathbf{b},\mathbf{c}$ are in A.P., then $\mathbf{2}\mathbf{b}=\mathbf{a}+\mathbf{c}$.

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Ex. 15 Check whether $a_n=2n^2+1$ is an A.p. or not.

ARITHMATIC PROGRESSIONS

Sol. $a_n=2n^2+1$ Then $a_{n+1}=2(n+1{)}^2+1$

$\begin{array}{rl}\therefore a_{n+1}-a_n& =2(n^2+2n+1)+1-2n^2-1\\ & =2n^2+4n+2+1-2n^2-1\\ & =4n+2,\text{ which is not constant }\therefore \text{ The above sequence is not an A.P. }\end{array}$

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DAILY PRACTIVE PTOBLEMS 6

OBJECTIVE DPP - 6.1

1. $p^{\text{th }}$ term of the series $(3-\frac{1}{n})+(3+\frac{2}{n})+(3-\frac{3}{n})+$ will be

(A) $3+\frac{p}{n}$

(B) $3-\frac{p}{n}$

(C) $3+\frac{n}{p}$

(D) $3-\frac{n}{p}$

ARITHMATIC PROGRESSIONS

ARITHMATIC PROGRESSIONS

2. $8^{\text{th }}$ term of the series $2\sqrt{2}+\sqrt{2}+0+\dots$ will be

(A) $-5\sqrt{2}$

(B) $5\sqrt{2}$

(C) $10\sqrt{2}$

(D) $-10\sqrt{2}$

ARITHMATIC PROGRESSIONS

ARITHMATIC PROGRESSIONS

3. If $9^{\text{th }}$ term of an A.P. be zero then the ratio of its ${29}^{\text{th }}$ and ${19}^{\text{th }}$ term is

(A) $1:2$

(B) $2:1$

(C) $1:3$

(D) $3:1$

ARITHMATIC PROGRESSIONS

ARITHMATIC PROGRESSIONS

4. Which term of the sequence $3,8,13,18,\dots$. is 498

(A) ${95}^{\text{th }}$

(B) 100 th

(C) ${102}^{\text{th }}$

(D) ${101}^{\text{th }}$

ARITHMATIC PROGRESSIONS

ARITHMATIC PROGRESSIONS

5. Which of the following sequence is an A.P.

(A) $f(n)=ab+bn\in N$

(B) $f(n)=k{r}^n,n\in N$

(C) $f(n)=(an+b)k{r}^n,n\in N$

(D) $f(n)=\frac{1}{a(n+\frac{b}{n})},n\in N$

ARITHMATIC PROGRESSIONS

ARITHMATIC PROGRESSIONS

6. If the $n^{\text{th }}$ term of an A.P. be $(2n-1)$ then the sum of its firs $n$ terms will be

(A) $n^2-1$

(B) $(2n-1{)}^2$

(C) $n^2$

(D) $n^2+1$

ARITHMATIC PROGRESSIONS

ARITHMATIC PROGRESSIONS

7. The interior angles of polygon are in A.P. if the smallest angles be ${120}^0$ and the common difference be 5 , then the number of sides is

(A) 8

(B) 10

(C) 9

(D) 6

ARITHMATIC PROGRESSIONS

ARITHMATIC PROGRESSIONS

8. In the first, second and last terms of an A.P. be $a,b,2a$ respectively, then its sum will

(A) $\frac{ab}{-a+b}$

(B) $\frac{ab}{2(b-a)}$

(C) $\frac{3ab}{2(b-a)}$

(D) $\frac{3ab}{4(b-a)}$

ARITHMATIC PROGRESSIONS

ARITHMATIC PROGRESSIONS

SUBJECTIVE DPP - 6.2

1. Is 51 a term of the A.P. $5,8,11,14,\dots \dots .$. ?

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Sol. 1 $No$

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2. Find the common difference of an A.P. whose first term is 100 and the sum of whose first six terms is five times the sum of the next six terms.

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Sol. 2 $-10$

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3. Find three number in A.P. whose sum is 21 and their product is 336.

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Sol. 3 $6,7,8$

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4. A student purchased a pen for Rs. 100. At the end of 8 years, it was valued at Rs. 20. Assuming the yearly depreciation is constant amount, find the annual depreciation.

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Sol. 4 $10$

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5. The fourth term of an A.P. is equal to three times the first term and the seventh term exceeds twice the third by one. Find the first term and the common difference.

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Sol. 5 $3,2$

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6. Which term of the sequence $17,16\frac{1}{5},15\frac{2}{5},14\frac{3}{5}\dots \dots$ is the first negative term.

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Sol. 6 ${23}^{rd}$

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7. If $S_n=n^{2}p$ and $S_m=m^{2}p(m\ne n)$ in an A.P. Prove that $S_p=p^3$.

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8. Find the sum of all the three digit numbers which leave remainder 2 when divided by 5 .

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Sol. 8 $98910$

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9. Find the sum of all two digit odd positive numbers

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Sol. 9 $2475$

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10. Find the ${10}^{\text{th }}$ term from end of the A.P. 4,9,14,….. 254 .

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Sol. 10 $209$

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11. 200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on. In how many rows the 200 logs are placed and how many logs are in the top row ?

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Sol. 11 $16$ rows, $5$ logs

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12. The sum of the first $n$ term of an A.P. is given by $S_n=3n^2-4n$. Determine the A.P. and its ${12}^{\text{th }}$ term.

[CBSE - 2004]

ARITHMATIC PROGRESSIONS

Sol. 12 $-1,5,11,\dots ..$& $a_{12}=65$

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13. Find the sum of the first 25 terms of an A.P. whose $n^{\text{th }}$ term is given by $t_n=2-3n$

[CBSE - 2004]

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Sol. 13 $-925$

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14. Find the number of terms of A.P. 54, 54, 48….. so that their sum is 513 .

[CBSE - 2005]

ARITHMATIC PROGRESSIONS

Sol. 14 $18,19$

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15. In an A.P., the sum of first $n$ terms is $\frac{3n^2}{2}+\frac{5n}{2}$ Find its ${25}^{\text{th }}$ term.

[CBSE - 2006]

ARITHMATIC PROGRESSIONS

Sol. 15 $76$

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16. Which term of the arithmetic progression $8,1420,26$,…. will be 72 more than its ${41}^{\text{st }}$ term ?

[CBSE - 2006]

ARITHMATIC PROGRESSIONS

Sol. 16 ${53}^{rd}$

ARITHMATIC PROGRESSIONS

17. The first term, common difference and last term of an A.P. are 12, 6 and 252 respectively. Find the sum of all terms of this A.P.

[CBSE - 2007]

ARITHMATIC PROGRESSIONS

Sol. 17 $5412$

ARITHMATIC PROGRESSIONS

18. Write the next term of the $\sqrt{8},\sqrt{18},\sqrt{32}$,

[CBSE - 2008]

ARITHMATIC PROGRESSIONS

Sol. 18 $\sqrt{50}$

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19. The sum of the $4^{\text{th }}$ and $8^{\text{th }}$ terms of an A.P. is 24 and the sum of the $6^{\text{th }}$ and ${10}^{\text{th }}$ terms is 44 . Find the first three terms of the A.P.

[CBSE - 2008]

ARITHMATIC PROGRESSIONS

Sol. 19 $-13,-8,-3$