If {S₁},{S₂},{S₃},...........{Sₘ} are sums of n terms of m A.P.'s whose first terms are

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8. Sequences and Series

medium

If ${S_1},\;{S_2},\;{S_3},...........{S_m}$ are the sums of $n$ terms of $m$ $A.P.'s$ whose first terms are $1,\;2,\;3,\;...............,m$ and common differences are $1,\;3,\;5,\;...........2m - 1$ respectively, then ${S_1} + {S_2} + {S_3} + .......{S_m} = $

$\frac{1}{2}mn(mn + 1)$

$mn(m + 1)$

$\frac{1}{4}mn(mn - 1)$

None of the above

Solution

(a) Here $a = 1,\;2,\;3,\,……..,m;\;\;\;d = 1,\;3,\;5,……..,2m – 1$

and $n = n$, then ${S_1} + {S_2} + ……. + {S_m} = \frac{1}{2}mn(mn + 1)$

$\left[ {{\rm{Using}}\;S\; = \frac{m}{2}(a + l).\;{\rm{Since}}\;{S_1},\;{S_2},\;{S_3},……{S_m}\;{\rm{form}}\;{\rm{an}}\;{\rm{A}}{\rm{.P}}{\rm{.}}} \right]$

Standard 11

Mathematics

If $3^{2 \sin 2 \alpha-1},14$ and $3^{4-2 \sin 2 \alpha}$ are the first three terms of an $A.P.$ for some $\alpha$, then the sixth term of this $A.P.$ is

hard

(JEE MAIN-2020)

Let ${a_1},{a_2},\;.\;.\;.\;.,{a_{49}}$ be in $A.P.$ such that $\mathop \sum \limits_{k = 0}^{12} {a_{4k + 1}} = 416$ and ${a_9} + {a_{43}} = 66$. If $a_1^2 + a_2^2 + \ldots + a_{17}^2 = 140m,$ then $m = \;\;..\;.\;.\;.\;$

hard

(JEE MAIN-2018)

Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic progression. If $S_{3 n}=3 S_{2 n}$, then the value of $\frac{S_{4 n}}{S_{2 n}}$ is:

medium

(JEE MAIN-2021)

If in the equation $a{x^2} + bx + c = 0,$ the sum of roots is equal to sum of square of their reciprocals, then $\frac{c}{a},\frac{a}{b},\frac{b}{c}$ are in

hard

Insert $6$ numbers between $3$ and $24$ such that the resulting sequence is an $A.P.$

medium

If ${S_1},\;{S_2},\;{S_3},...........{S_m}$ are the sums of $n$ terms of $m$ $A.P.'s$ whose first terms are $1,\;2,\;3,\;...............,m$ and common differences are $1,\;3,\;5,\;...........2m - 1$ respectively, then ${S_1} + {S_2} + {S_3} + .......{S_m} = $

Solution

Similar Questions

If $3^{2 \sin 2 \alpha-1},14$ and $3^{4-2 \sin 2 \alpha}$ are the first three terms of an $A.P.$ for some $\alpha$, then the sixth term of this $A.P.$ is

Let ${a_1},{a_2},\;.\;.\;.\;.,{a_{49}}$ be in $A.P.$ such that $\mathop \sum \limits_{k = 0}^{12} {a_{4k + 1}} = 416$ and ${a_9} + {a_{43}} = 66$. If $a_1^2 + a_2^2 + \ldots + a_{17}^2 = 140m,$ then $m = \;\;..\;.\;.\;.\;$

Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic progression. If $S_{3 n}=3 S_{2 n}$, then the value of $\frac{S_{4 n}}{S_{2 n}}$ is:

If in the equation $a{x^2} + bx + c = 0,$ the sum of roots is equal to sum of square of their reciprocals, then $\frac{c}{a},\frac{a}{b},\frac{b}{c}$ are in

Insert $6$ numbers between $3$ and $24$ such that the resulting sequence is an $A.P.$

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