The sum of all terms of the n^{th} bracket of | vedclass

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

Question

(D) The number of terms in successive groups are $1, 2, 3, \dots, n$. Thus,the $n^{th}$ group contains $n$ terms in an $A$.$P$. with a common differen

Vedclass · Accepted Answer

$n^3$

Vedclass · Answer

(D) The number of terms in successive groups are $1, 2, 3, \dots, n$. Thus,the $n^{th}$ group contains $n$ terms in an $A$.$P$. with a common difference of $d = 2$.First,we find the first term of the $n^{th}$ group. The first terms of the groups are $1, 3, 7, 13, \dots$. Let $a_n$ be the first term of the $n^{th}$ group. The differences between consecutive first terms are $2, 4, 6, \dots$,which form an $A$.$P$.$a_n = a_1 + \sum_{k=1}^{n-1} (2k) = 1 + 2 \cdot \frac{(n-1)n}{2} = 1 + n^2 - n = n^2 - n + 1$.The $n^{th}$ group is an $A$.$P$. with $n$ terms,first term $a = n^2 - n + 1$,and common difference $d = 2$.The sum of the $n^{th}$ group is $S_n = \frac{n}{2} [2a + (n-1)d]$.$S_n = \frac{n}{2} [2(n^2 - n + 1) + (n-1)2]$$S_n = \frac{n}{2} [2n^2 - 2n + 2 + 2n - 2]$$S_n = \frac{n}{2} [2n^2] = n^3$.

The sum of all terms of the $n^{th}$ bracket of the sequence $(1), (3, 5), (7, 9, 11), \dots$ is equal to:

Similar Questions

If $\frac{S_n}{S_m} = \frac{n^4}{m^4}$ (where $S_k$ is the sum of the first $k$ terms of an $A$.$P$. $a_1, a_2, \dots, \infty$),then the value of $\frac{a_{m+1}}{a_{n+1}}$ in terms of $m$ and $n$ will be

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

If $n$ arithmetic means $a_1, a_2, \dots, a_n$ are inserted between $50$ and $100$ and $n$ harmonic means $h_1, h_2, \dots, h_n$ are inserted between the same two numbers,then $a_2 h_{n-1}$ is equal to

If the sum of the roots of the equation $ax^2 + bx + c = 0$ is equal to the sum of the reciprocals of their squares,then $bc^2, ca^2, ab^2$ will be in

The sum of the series $1 + \frac{4}{3} + \frac{10}{9} + \frac{28}{27} + \dots$ up to $n$ terms is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module