(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

Mathematics Questions

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Sequences and Series

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Arithmetic progression

The ratio of the sum of $m$ and $n$ terms of an $A.P.$ is $m^2:n^2$. Then the ratio of the $m^{th}$ and $n^{th}$ term will be:

Medium

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If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of an arithmetic sequence are $a$,$b$,and $c$ respectively,then the value of $[a(q - r) + b(r - p) + c(p - q)]$ is:

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Let $9 < x_1 < x_2 < \ldots < x_7$ be in an $A.P.$ with common difference $d$. If the standard deviation of $x_1, x_2, \ldots, x_7$ is $4$ and the mean is $\overline{x}$,then $\overline{x} + x_6$ is equal to:

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Which of the following sequences is an arithmetic sequence?

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If $a_1, a_2, ..., a_{24}$ are in an arithmetic progression and $a_1 + a_5 + a_{10} + a_{15} + a_{20} + a_{24} = 225$,then what is the sum of the first $24$ terms of this arithmetic progression?

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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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