Question

If an denotes the nth term of an A.P. and Sn denotes the sum of its n terms, and
S1= 6, S7 = 105, show that an: an-2 = (n + 1): (n - 1).​

Answer 1

aₙ : aₙ₋₂  = (n + 1)  :  (n - 1)

Step-by-step explanation:

aₙ  = nth term

Sₙ = Sum of nth term

S₁ = a₁

S₁  = 6

=> a₁ = 6

S₇ = (7/2)(a₁  + a₁ + 6d)

=> (7/2)(2a₁ + 6d)  = 105

=> a₁ + 3d  = 15

=> 6 + 3d = 15

=> 3d = 9

=> d = 3

aₙ = a₁ + (n-1)d  =  6 + (n-1)3  = 6 + 3n - 3  = 3n + 3

aₙ₋₂ = a₁ + (n-2-1)d  =  6 + (n-3)3  = 6 + 3n - 9  = 3n - 3

aₙ : aₙ₋₂  = (3n + 3)  :  (3n - 3)

=> aₙ : aₙ₋₂  = (n + 1)  :  (n - 1)

QED

proved

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