If a₁ , a₂, a₃, . . . . , aₙ, .... are in A.P. such that a₄ - ag + a_{10}, = m , then sum

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8. Sequences and Series

hard

If $a_1 , a_2, a_3, . . . . , a_n, ....$ are in $A.P.$ such that $a_4 - a_7 + a_{10}\, = m$, then the sum of first $13$ terms of this $A.P.$, is .............. $\mathrm{m}$

A

$10$

B

$12$

C

$13$

D

$15$

(JEE MAIN-2013)

Solution

If $d$ be the common differnce, then

$m = {a_4} – {a_7} + {a_{10}} = {a_4} – {a_7} + {a_7} + 3d = {a_7}$

${S_{13}} = \frac{{13}}{2}\left[ {{a_1} + {a_{13}}} \right] = \frac{{13}}{2}\left[ {{a_1} + {a_7} + 6d} \right]$

$\,\,\,\,\,\,\, = \frac{{13}}{2}\left[ {2{a_7}} \right] = 13{a_7} = 13\,\,m$

Standard 11

Mathematics

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