(B) Given: First term $= a$,Last term $= b$,Common difference $d = 1$.The formula for the $n^{th}$ term is $T_n = a + (n - 1)d$.Substituting the value

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(B) Given: First term $= a$,Last term $= b$,Common difference $d = 1$.The formula for the $n^{th}$ term is $T_n = a + (n - 1)d$.Substituting the values,we get $b = a + (n - 1)(1)$.$b - a = n - 1$,which implies $n = b - a + 1$.The sum of an arithmetic progression is given by $S_n = \frac{n}{2}(a + l)$,where $l$ is the last term.Substituting $n = b - a + 1$ and $l = b$:$S_n = \frac{(b - a + 1)(a + b)}{2}$.

Class 11 Mathematics Sequences and Series Arithmetic progression

Medium

If the first term of an arithmetic progression is $a$,the common difference is $1$,and the last term is $b$,what is the sum of the progression?

A
$\frac{(a + b)(1 + a - b)}{2}$
B
$\frac{(a + b)(1 - a + b)}{2}$
C
$\frac{(a + b)(1 - a)}{2}$
D
$(a + b)(1 - a + b)$

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Class 11

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Sequences and Series

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Arithmetic progression

Let $a_1, a_2, a_3, \ldots$ be in an $A.P.$ such that $\sum_{k=1}^{12} a_{2k-1} = -\frac{72}{5} a_1$,where $a_1 \neq 0$. If $\sum_{k=1}^{n} a_k = 0$,then $n$ is:

MediumJEE MAIN 2025

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If $a$,$b$,and $c$ are the first,second,and last terms of an arithmetic progression respectively,then the total number of terms is......

Medium

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There are $n$ arithmetic means between $7$ and $71$. If the $5^{th}$ arithmetic mean is $27$,then $n = ......$

Easy

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The number of terms of the $A.P. 3, 7, 11, 15, ...$ to be taken so that the sum is $406$ is

Easy

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The interior angles of a polygon with $n$ sides are in an $A.P.$ with a common difference of $6^{\circ}$. If the largest interior angle of the polygon is $219^{\circ}$,then $n$ is equal to . . . . . .

DifficultJEE MAIN 2025

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If the first term of an arithmetic progression is $a$,the common difference is $1$,and the last term is $b$,what is the sum of the progression?

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Let $a_1, a_2, a_3, \ldots$ be in an $A.P.$ such that $\sum_{k=1}^{12} a_{2k-1} = -\frac{72}{5} a_1$,where $a_1 \neq 0$. If $\sum_{k=1}^{n} a_k = 0$,then $n$ is:

If $a$,$b$,and $c$ are the first,second,and last terms of an arithmetic progression respectively,then the total number of terms is......

There are $n$ arithmetic means between $7$ and $71$. If the $5^{th}$ arithmetic mean is $27$,then $n = ......$

The number of terms of the $A.P. 3, 7, 11, 15, ...$ to be taken so that the sum is $406$ is

The interior angles of a polygon with $n$ sides are in an $A.P.$ with a common difference of $6^{\circ}$. If the largest interior angle of the polygon is $219^{\circ}$,then $n$ is equal to . . . . . .

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