Let a_1, a_2, a_3, ldots .. be a sequence of | vedclass

Name: PDF Generator App | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

Given $2\left( a _1+ a _2+\ldots . .+ a _{ n }ight)= b _1+ b _2+\ldots . .+ b _{ n }$

$\Rightarrow \quad 2 	imes \frac{ n }{2}\left(2 c +( n -2)

Vedclass · Accepted Answer

$1$

Vedclass · Answer

Given $2\left( a _1+ a _2+\ldots . .+ a _{ n }ight)= b _1+ b _2+\ldots . .+ b _{ n }$

$\Rightarrow \quad 2 	imes \frac{ n }{2}\left(2 c +( n -2) x _2ight)= c \left(\frac{2^{ n }-1}{2-1}ight)$

$\Rightarrow \quad 2 n ^2-2 n = c \left(2^{ n }-1-2 n ight)$

$\Rightarrow \quad c =\frac{2 n ^2-2 n }{2^{ n }-1-2 n } \in N$

$	ext { So, }  2 n ^2-2 n \geq 2^{ n }-1-2 n$

$\Rightarrow \quad  2 n ^2+1 \geq 2^{ n } \Rightarrow n <7$

$\Rightarrow \quad  n 	ext { can be } 1,2,3, \ldots.$

Checking $c$ against these values of $n$

we get $c=12 \quad($ when $n=3)$

Hence number of such $c=1$

Similar Questions

If the sum of first 6 term is $9$ times to the sum of first $3$ terms of the same $G.P.$, then the common ratio of the series will be

The two geometric means between the number $1$ and $64$ are

Find the sum up to $20$ terms in the geometric progression $0.15,0.015,0.0015........$

Let $S_1$ be the sum of areas of the squares whose sides are parallel to coordinate axes. Let $S_2$ be the sum of areas of the slanted squares as shown in the figure. Then, $\frac{S_1}{S_2}$ is equal to

Find the sum of the products of the corresponding terms of the sequences $2,4,8,16,32$ and $128,32,8,2, \frac{1}{2}$