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Find the common difference of an AP, whose first term is 5 and the sum of its first four terms is half the sum of the next four terms - Mathematics

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Sum

Find the common difference of an AP, whose first term is 5 and the sum of its first four terms is half the sum of the next four terms

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Solution

`a_1 + a_2 + a_3 + a_4 = (a_5 + a_6 + a_7 + a_8 )`

`⇒ 2[a_1 + a_2 + a_3 + a_4 ] = a_5 + a_6 + a_7 + a_8`

`⇒ 2[a_1 + a_2 + a_3 + a_4 ] + (a_1 + a_2 + a_3 + a_4 ) = [a_1 + a_2 + a_3 + a_4 ]+ (a_5 + a_6 + a_7 + a_8 )`

(adding both side `a_1 + a_2 + a_3 + a_4`)

`⇒ 3(a_1 + a_2 + a_3 + a_4 ) = a_1 + …. + a_8 ⇒ 3S_4 = S_8`

`=>3[4/2(2xx5+(4-1)d)]=[8/2(2xx5+(8-1)d)]`

⇒ 3[10 + 3d] = 2[10 + 7d]

⇒ 30 + 9d = 20 + 14d ⇒ 5d = 10 ⇒ d = 2

Concept: Arithmetic Progression
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