Statement-I: If the sum of n terms of a seque | vedclass

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

Question

(D) For a sequence to be an $AP$,the sum of $n$ terms $S_n$ must be of the form $An^2 + Bn$.In Statement-$I$,$S_n = 6n^2 + 3n + 1$. Since there is a c

Vedclass · Accepted Answer

Statement-$I$ is false. Statement-$II$ is true.

Vedclass · Answer

(D) For a sequence to be an $AP$,the sum of $n$ terms $S_n$ must be of the form $An^2 + Bn$.In Statement-$I$,$S_n = 6n^2 + 3n + 1$. Since there is a constant term $(1)$,this cannot be an $AP$. Thus,Statement-$I$ is false.In Statement-$II$,the sum of $n$ terms of an $AP$ with first term $a$ and common difference $d$ is given by $S_n = \frac{n}{2}[2a + (n-1)d] = \frac{d}{2}n^2 + (a - \frac{d}{2})n$. This is clearly of the form $an^2 + bn$. Thus,Statement-$II$ is true.Therefore,Statement-$I$ is false and Statement-$II$ is true.

Statement-$I$: If the sum of $n$ terms of a sequence is $6n^2 + 3n + 1$,then it is an Arithmetic Progression $(AP)$.
Statement-$II$: The sum of $n$ terms of an Arithmetic Progression is always in the form $an^2 + bn$.

Similar Questions

Let $a_{1}, a_{2}, \ldots, a_{n}$ be a given $A.P.$ whose common difference is an integer and $S_{n} = a_{1} + a_{2} + \ldots + a_{n}$. If $a_{1} = 1$,$a_{n} = 300$ and $15 \leq n \leq 50$,then the ordered pair $(S_{n-4}, a_{n-4})$ is equal to:

If $1, \log_9(3^{1-x} + 2), \log_3(4 \cdot 3^x - 1)$ are in an arithmetic progression,then $x = \dots$

Write the first five terms of the sequence whose $n^{th}$ term is $a_{n} = \frac{2n - 3}{6}$.

The first term of an $A.P.$ of consecutive integers is ${p^2} + 1$. The sum of $(2p + 1)$ terms of this series can be expressed as:

Let $A = \{1, a_{1}, a_{2}, \ldots, a_{18}, 77\}$ be a set of integers with $1 < a_{1} < a_{2} < \ldots < a_{18} < 77$. Let the set $A + A = \{x + y : x, y \in A\}$ contain exactly $39$ elements. Then,the value of $a_{1} + a_{2} + \ldots + a_{18}$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Statement-$I$: If the sum of $n$ terms of a sequence is $6n^2 + 3n + 1$,then it is an Arithmetic Progression $(AP)$.Statement-$II$: The sum of $n$ terms of an Arithmetic Progression is always in the form $an^2 + bn$.