Question

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Let a1, a2, a3, …… , a100 be an arithmetic progression with a1 = 3 and SP = Σai, where the summation runs over p and 1 ≤ p ≤ 100. For any integer n with 1 ≤ n ≤ 20, let m = 5n. What is the value of a2 if Sm/ Sndoes not depend on n.

Answer 1

Answer:

a₂ = 3  or 9

Step-by-step explanation:

Sm  = (m/2)(2a + (m - 1)d)

Sm = (5n/2)(2*3 + (5n-1)d)

Sm = (5n/2) ( 6 - d + 5nd)

Sn  = (n/2)(2a + (n - 1)d)

Sn= (n/2)(6 -d  + nd )

Sm /Sn  =  (5n/2) ( 6 - d + 5nd) / ((n/2)(6 -d  + nd )

= 5 (6 - d  + 5nd) /(6 - d + nd)

Let say  5 (6 - d  + 5nd) /(6 - d + nd) = k

=> 30 - 5d + 25nd = 6k - kd + knd

=> nd(25 - k) = d(5- k) - 6(5 - k)

=> nd(25 - k)  = (d - 6)(5 - k)

as these are independnt of n  , hence

d = 0 , k = 25  , d = 6 , k = 5 are the solutions

d = 0  , d = 6

a₂ = a₁ + d

a₁ = 3

a₂ = 3 + 0  or 3 + 6

a₂ = 3  or 9

Answer 2

Step-by-step explanation:

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