Let { a n } and { b n } be two different sequences such that a n + 1 = 2 + a n n N , a 10 = 21 and b n = 1 a n a n + 2 n N , then the value of 15 r = 1 b r is

Question

Let ${a_{n}}$ and ${b_{n}}$ be two different sequences such that $a_{n + 1} = 2 + a_{n}$ $\forall n \in N,$ $a_{10} = 21$ and $b_{n} = \frac{1}{a_{n} a_{n + 2}}$ $\forall n \in N$ , then the value of $15 \sum_{r = 1}^{\infty} b_{r}$ is

Smart Achievers · Accepted Answer

(2)

$a_{1}, a_{2}, a_{3}, \dots are in A.P.$

$d = 2, a_{10} = 21 \Rightarrow a_{1} = 3, a_{n} = 2 n + 1$

$\sum_{r = 1}^{\infty} b_{r} = \sum_{n = 1}^{\infty} \frac{1}{(2 n + 1) (2 n + 5)}$

$= \frac{1}{4} \sum_{n = 1}^{\infty} (\frac{1}{2 n + 1} - \frac{1}{2 n + 5})$

$= \frac{1}{4} [\frac{1}{3} + \frac{1}{5}]$

$= \frac{2}{15}$

Solution

Q.
Let ${a_{n}}$ and ${b_{n}}$ be two different sequences such that $a_{n + 1} = 2 + a_{n}$ $\forall n \in N,$ $a_{10} = 21$ and $b_{n} = \frac{1}{a_{n} a_{n + 2}}$ $\forall n \in N$ , then the value of $15 \sum_{r = 1}^{\infty} b_{r}$ is

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Solution

Q. Let {an} and {bn} be two different sequences such that an+1=2+an ∀n∈N, a10=21 and bn=1anan+2 ∀n∈N, then the value of 15∑r=1∞br is

Ans. (2) a1,a2,a3,… are in A.P. d=2, a10=21⇒a1=3, an=2n+1 ∑r=1∞br=∑n=1∞1(2n+1)(2n+5) =14∑n=1∞(12n+1-12n+5) =14[13+15] =215

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Q.
Let ${a_{n}}$ and ${b_{n}}$ be two different sequences such that $a_{n + 1} = 2 + a_{n}$ $\forall n \in N,$ $a_{10} = 21$ and $b_{n} = \frac{1}{a_{n} a_{n + 2}}$ $\forall n \in N$ , then the value of $15 \sum_{r = 1}^{\infty} b_{r}$ is