For a sequence- - r -1100 a r - - r -150 a 2 r -1- - F ind - r -150 a 2 r

Α = ^100_r=1 a_r = =a_1+a_2+a_3——–+a_98+a_99+a_100 β = ^50_r=1 a_2r 1 = a_1+a_3——–+a_99 Let r = ^50_r=1 a_2r = a_2+a_4——–+a_100 Consider α β = a_2+a_4——–+a_1 ...

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Byju's Answer

Standard XII

Mathematics

Summation by Sigma Method

For a sequenc...

Question

For a sequence, $Σ_{r = 1}^{100} a_{r} = α, Σ_{r = 1}^{50} a_{2 r - 1} = β . F i n d Σ_{r = 1}^{50} a_{2 r}$

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Solution

$α$ = $Σ_{r = 1}^{100} a_{r} =$ = $a_{1} + a_{2} + a_{3} - - - - - - - - + a_{98} + a_{99} + a_{100}$
$β$ = $Σ_{r = 1}^{50} a_{2 r - 1}$ = $a_{1} + a_{3} - - - - - - - - + a_{99}$
Let r = $Σ_{r = 1}^{50} a_{2 r}$ = $a_{2} + a_{4} - - - - - - - - + a_{100}$

Consider
$α - β$ = $a_{2} + a_{4} - - - - - - - - + a_{100}$
=r
$\Rightarrow Σ_{r = 1}^{50} a_{2 r} = α - β$

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