The natural numbers are arranged in rows as f | vedclass

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

Question

(B) The number of terms in the $n^{th}$ row is $n$.The last term of the $(n-1)^{th}$ row is the sum of the first $(n-1)$ natural numbers,which is $\fr

Vedclass · Accepted Answer

$\frac{n}{2}(n^2 + 1)$

Vedclass · Answer

(B) The number of terms in the $n^{th}$ row is $n$.The last term of the $(n-1)^{th}$ row is the sum of the first $(n-1)$ natural numbers,which is $\frac{(n-1)n}{2}$.Therefore,the first term of the $n^{th}$ row is $\frac{(n-1)n}{2} + 1 = \frac{n^2 - n + 2}{2}$.The terms in the $n^{th}$ row form an arithmetic progression $(A.P.)$ with $n$ terms,first term $a = \frac{n^2 - n + 2}{2}$,and common difference $d = 1$.The sum of the $n$ terms is given by $S_n = \frac{n}{2}[2a + (n-1)d]$.Substituting the values: $S_n = \frac{n}{2}[2(\frac{n^2 - n + 2}{2}) + (n-1)(1)]$.$S_n = \frac{n}{2}[n^2 - n + 2 + n - 1] = \frac{n}{2}(n^2 + 1)$.

The natural numbers are arranged in rows as follows:
$1$
$2, 3$
$4, 5, 6$
$7, 8, 9, 10$
$. . .$
What is the sum of the numbers in the $n^{th}$ row?

Similar Questions

If $a, b, c$ are in Arithmetic Progression,then $(a - c)^2 = \dots$

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 $an^2 + bn$.

The sum of the first and third term of an arithmetic progression is $12$ and the product of the first and second term is $24$. Find the first term.

The sum of three terms of an Arithmetic Progression $(AP)$ is $18$ and the sum of their squares is $158$. The largest term is.......

If the sum of the first $11$ terms of an $A.P.$,$a_{1}, a_{2}, a_{3}, \ldots$ is $0$ $(a_{1} \neq 0)$,then the sum of the $A.P.$,$a_{1}, a_{3}, a_{5}, \ldots, a_{23}$ is $k a_{1}$,where $k$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module

The natural numbers are arranged in rows as follows:$1$$2, 3$$4, 5, 6$$7, 8, 9, 10$$. . .$What is the sum of the numbers in the $n^{th}$ row?