If x_n = frac{2n^2 + n + 1}{2n^2 - 3n + 2},th | vedclass

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

Question

(B) Given $x_n = \frac{2n^2 + n + 1}{2n^2 - 3n + 2}$. Let $f(n) = 2n^2 - 3n + 2$. Then $f(n-1) = 2(n-1)^2 - 3(n-1) + 2 = 2(n^2 - 2n + 1) - 3n + 3 + 2

Vedclass · Accepted Answer

$\frac{n(n + 3)}{2}$

Vedclass · Answer

(B) Given $x_n = \frac{2n^2 + n + 1}{2n^2 - 3n + 2}$. Let $f(n) = 2n^2 - 3n + 2$. Then $f(n-1) = 2(n-1)^2 - 3(n-1) + 2 = 2(n^2 - 2n + 1) - 3n + 3 + 2 = 2n^2 - 7n + 7$. This does not match the numerator directly. Let us re-evaluate $x_n = \frac{2n^2 + n + 1}{2(n-1)^2 - 3(n-1) + 2} = \frac{2n^2 + n + 1}{2n^2 - 7n + 7}$.Actually,observing the product $\prod_{i=1}^r x_i$: for $r=1$,$x_1 = \frac{2+1+1}{2-3+2} = \frac{4}{1} = 4$. For $r=2$,$x_1 x_2 = 4 	imes \frac{8+2+1}{8-6+2} = 4 	imes \frac{11}{4} = 11$. For $r=3$,$x_1 x_2 x_3 = 11 	imes \frac{18+3+1}{18-9+2} = 11 	imes \frac{22}{11} = 22$. The product is $2r^2 + r + 1$.The second term is $2 \sum_{i=1}^r (2i - 1) = 2(r^2) = 2r^2$.Thus,the expression inside the summation is $(2r^2 + r + 1) - 2r^2 = r + 1$.Finally,$\sum_{r=1}^n (r + 1) = \sum_{r=1}^n r + \sum_{r=1}^n 1 = \frac{n(n+1)}{2} + n = \frac{n^2 + n + 2n}{2} = \frac{n(n+3)}{2}$.

If $x_n = \frac{2n^2 + n + 1}{2n^2 - 3n + 2}$,then $\sum_{r=1}^n \left[ \left( \prod_{i=1}^r x_i \right) - 2\sum_{i=1}^r (2i - 1) \right]$ is equal to

Similar Questions

Let $A$ be the sum of the first $20$ terms and $B$ be the sum of the first $40$ terms of the series $1^2 + 2 \cdot 2^2 + 3^2 + 2 \cdot 4^2 + 5^2 + \dots$. If $B - 2A = 100\lambda$,then $\lambda$ is equal to:

For all positive integral values of $n$,the value of $3 \cdot 1 \cdot 2 + 3 \cdot 2 \cdot 3 + 3 \cdot 3 \cdot 4 + \dots + 3 \cdot n \cdot (n + 1)$ is

What is the arithmetic mean of the first $n$ terms of the sequence $1 \times 3 \times 5, 3 \times 5 \times 7, 5 \times 7 \times 9, \dots$?

The value of $1^3 - 2^3 + 3^3 - \dots + 15^3$ is:

Find the sum to $n$ terms of the series $1 \times 2 + 2 \times 3 + 3 \times 4 + 4 \times 5 + \ldots$

Vedclass Test Series

Exam Paper Generator

Online Exam Module