SATHEE: Sequences and Series 2 Question 5

Sequences and Series 2 Question 5

6. $n_{1}, n_{2}, \dots, n_{p}$ are $p$ positive integers, whose sum is an even number, then the number of odd integers among them is odd.

(1985, 1M)

Integer Answer Type Question

Show Answer

Answer:

Correct Answer: 6. (a)

Solution:

Clearly, the two digit number which leaves remainder 2 when divided by 7 is of the form $N = 7 k + 2$ [by Division Algorithm]

For,

$\begin{matrix} k = 2, N = 16 \\ k = 3, N = 23 \\ ⋮ \\ k = 13, N = 93 \end{matrix}$

$∴ 12$ such numbers are possible and these numbers forms an AP.

Now,

$\begin{aligned} S = \frac{12}{2} [16 + 93] = & 654 \\ ∵ S_{n} = & \frac{n}{2} (a + l) \end{aligned}$

Similarly, the two digit number which leaves remainder 5 when divided by 7 is of the form $N = 7 k + 5$

For $k = 1, N = 12$

$\begin{aligned} k = 2, N = 19 \\ ⋮ \\ k = 13, N = 96 \end{aligned}$

$∴ 13$ such numbers are possible and these numbers also forms an AP.

Now,

$\begin{aligned} S^{'} = \frac{13}{2} [12 + 96] & = 702 \\ ∵ S_{n} = \frac{n}{2} (a + l) \end{aligned}$

Total sum $= S + S^{'} = 654 + 702 = 1356$

Sequences and Series 2 Question 5

6. $n_{1}, n_{2}, \dots, n_{p}$ are $p$ positive integers, whose sum is an even number, then the number of odd integers among them is odd.

Integer Answer Type Question

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Sequences and Series 2 Question 5

6. n1,n2,…,np are p positive integers, whose sum is an even number, then the number of odd integers among them is odd.

Integer Answer Type Question

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

6. $n_{1}, n_{2}, \dots, n_{p}$ are $p$ positive integers, whose sum is an even number, then the number of odd integers among them is odd.