Let a n be the nth term of an AP- If - r -150 a 2 r - and - r -150 a 2 r -1- then the common difference of the A-P

Another way of solving this is, ^50_r=1 a_2r = a_2+a_4+—–+a_100=α ^50_r=1 a_2r 1 = a_1+a_3+—–+a_99=β α β =(a_2 a_1)+(a_4 a_3)+(a_6 a_5)+——+(a_100 a_99) As com ...

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Byju's Answer

Standard XII

Mathematics

Arithmetic Progression

Let a n be th...

Question

Let $a_{n}$ be the nth term of an AP. If $Σ_{r = 1}^{50} a_{2 r} = α$ and $Σ_{r = 1}^{50} a_{2 r - 1} = β$ , then the common difference of the A.P

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Solution

Another way of solving this is,

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

$α - β = (a_{2} - a_{1}) + (a_{4} - a_{3}) + (a_{6} - a_{5}) + - - - - - - + (a_{100} - a_{99})$
As common difference 'd' is the difference between consecutive terms of A.P.

$α - β = d + d + - - - - - - + d (50 t i m e s)$
$\Rightarrow$ d= $\frac{α - β}{50}$

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Let $a_{n}$ be the $n^{t h}$ term of an AP. If $Σ_{r = 1}^{50} a_{2 r} = α$ and $Σ_{r = 1}^{50} a_{2 r - 1} = β$ , then what is the common difference of the A.P.