Let a n be a non - constant arithmetic progression with a 1-1 and for any n - 1- the valuea 2 n - a n-an-1- a1remains constant-Then- a 15 will be-A- 30B- 31C- 29D- Can-t be determined

The correct option is C 29C = a_2n+a_2n 1+a_2n 2+ ⋯ + a_n+1/a_n + a_n 1 + ⋯ + a_1 Adding 1 on both sides, we get C+ 1= a_2n+a_2n 1+a_2n 2+ ⋯ + a_n+1/a_n + a_n ...

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Byju's Answer

Standard XI

Mathematics

Selecting Consecutive Terms in A.P

Let a n be a ...

Question

Let ${a_{n}}$ be a non - constant arithmetic progression with $a_{1} = 1$ and for any $n \geq 1$ , the value

$\frac{a_{2 n} + a_{2 n - 1} + \dots + a_{n + 1}}{a_{n} + a_{n - 1} + \dots a_{1}}$
remains constant.

Then, $a_{15}$ will be?

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Can't be determined

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Solution

The correct option is B 29
$C = \frac{a_{2 n} + a_{2 n - 1} + a_{2 n - 2} + \dots + a_{n + 1}}{a_{n} + a_{n - 1} + \dots + a_{1}}$
Adding 1 on both sides, we get
$C + 1 = \frac{a_{2 n} + a_{2 n - 1} + a_{2 n - 2} + \dots + a_{n + 1}}{a_{n} + a_{n - 1} + \dots + a_{1}}$
$= \frac{a_{2 n} + a_{2 n - 1} + \dots + a_{n + 1} + a_{n} + \dots + a_{1}}{a_{n} + a_{n - 1} + \dots + a_{1}} ∴ C + 1 = \frac{S_{2 n}}{S_{n}} = \frac{\frac{2 n}{2} [2 a_{1} + (2 n - 1) d]}{\frac{n}{2} [2 a_{1} + (n - 1) d]} = \frac{2 [2 a_{1} + (2 n - 1) d]}{2 a_{1} + (n - 1) d}$
Since, $\frac{C + 1}{2}$ is a constant.
Let $\frac{C + 1}{2} = R$
$\Rightarrow \frac{2 + (2 n - 1) d}{2 + (n - 1) d} = R [∵ a_{1} = 1] \Rightarrow 2 + (2 n - 1) d = 2 R + (n - 1) d R \Rightarrow 2 R + n d R - d R = 2 + 2 n d - d \Rightarrow n d (R - 2) = d R - 2 R - d + 2 \Rightarrow n d (R - 2) = (d - 2) (R - 1)$
Now, left side has n which changes and right side remains constant
$∴$ LHS = RHS = 0
$\Rightarrow R = 2 [∵ n \neq 0, d \neq 0] \Rightarrow 0 = d - 2 \Rightarrow d = 2 ∴ a_{15} = a_{1} + (15 - 1) d = 1 + 14 \times 2 = 29$

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Let ${a_{n}}$ be a non - constant arithmetic progression with $a_{1} = 1$ and for any $n \geq 1$ , the value $\frac{a_{2 n} + a_{2 n - 1} + \dots + a_{n + 1}}{a_{n} + a_{n - 1} + \dots a_{1}}$ remains constant.

Then, $a_{15}$ will be?

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Let {an} be a non - constant arithmetic progression with a1=1 and for any n≥1, the value a2n+a2n−1+⋯+an+1an+an−1+⋯a1 remains constant. Then, a15 will be?

Let ${a_{n}}$ be a non - constant arithmetic progression with $a_{1} = 1$ and for any $n \geq 1$ , the value

$\frac{a_{2 n} + a_{2 n - 1} + \dots + a_{n + 1}}{a_{n} + a_{n - 1} + \dots a_{1}}$
remains constant.

Then, $a_{15}$ will be?