SATHEE: Sequences and Series 3 Question 1

Sequences and Series 3 Question 1

1. Let $a, b$ and $c$ be in GP with common ratio $r$ , where $a \neq 0$ and $0 < r \leq \frac{1}{2}$ . If $3 a, 7 b$ and $15 c$ are the first three terms of an AP, then the 4 th term of this AP is

(2019 Main, 10 April II)

(a) $5 a$

(b) $\frac{2}{3} a$

(d) $\frac{7}{3} a$

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Answer:

Correct Answer: 1. (c)

Solution:

Key Idea Use $n^{th}$ term of AP i.e., $a_{n} = a + (n - 1) d$ , If $a, A$ , $b$ are in AP, then $2 A = a + b$ and $n^{th}$ term of G.P. i.e., $a_{n} = a r^{n - 1}$ .

It is given that, the terms $a, b, c$ are in GP with common ratio $r$ , where $a \neq 0$ and $0 < r \leq \frac{1}{2}$ .

So, let, $b = a r$ and $c = a r^{2}$

Now, the terms $3 a, 7 b$ and $15 c$ are the first three terms of an AP, then

$\begin{aligned} 2 (7 b) = 3 a + 15 c \\ \begin{array}{lcr} \Rightarrow & 14 a r = 3 a + 15 a r^{2} & [as b = a r, c = a r^{2}] \\ \Rightarrow & 14 r = 3 + 15 r^{2} & [as a \neq 0] \end{array} \\ \Rightarrow 15 r^{2} - 14 r + 3 = 0 \\ \Rightarrow 15 r^{2} - 5 r - 9 r + 3 = 0 \\ \Rightarrow 5 r (3 r - 1) - 3 (3 r - 1) = 0 \\ \Rightarrow (3 r - 1) (5 r - 3) = 0 \\ \Rightarrow r = \frac{1}{3} or \frac{3}{5} \\ as, r \in 0, \frac{1}{2}, so r = \frac{1}{3} \end{aligned}$

Now, the common difference of $A P = 7 b - 3 a$

$= 7 a r - 3 a = a \frac{7}{3} - 3 = - \frac{2 a}{3}$

So, $4^{th}$ term of $A P = 3 a + 3 \frac{- 2 a}{3} = a$

Sequences and Series 3 Question 1

1. Let $a, b$ and $c$ be in GP with common ratio $r$ , where $a \neq 0$ and $0 < r \leq \frac{1}{2}$ . If $3 a, 7 b$ and $15 c$ are the first three terms of an AP, then the 4 th term of this AP is

Answer:

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Sequences and Series 3 Question 1

1. Let a,b and c be in GP with common ratio r, where a≠0 and 0<r≤12. If 3a,7b and 15c are the first three terms of an AP, then the 4 th term of this AP is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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1. Let $a, b$ and $c$ be in GP with common ratio $r$ , where $a \neq 0$ and $0 < r \leq \frac{1}{2}$ . If $3 a, 7 b$ and $15 c$ are the first three terms of an AP, then the 4 th term of this AP is