Question

a,b and c are the first, second and nth term of a G.P respectively.Express n and sum of the first n terms if the A.P in terms of a, b and c

Answer 1

Answer:

Step-by-step explanation:

a,b and c are the first, second and nth term of a G.P respectively.

=> First Term = a

second term = b

Common Ratio = b/a

nth Term=  First term * (common ratio)ⁿ⁻¹

=> c = a (b/a)ⁿ⁻¹

=> (b/a)ⁿ⁻¹  = c/a

=> (b/a)ⁿ = cb/a²

Taking log both sides

=> nlog(b/a)   = log(cb/a²)

=> n =  log(cb/a²) / log(b/a)

Sum

= a(rⁿ - 1)/(r - 1)

rⁿ = cb/a²    & r = $\sqrt[n]{cb/a^{2} }$

if AP

Then Common difference = b - a

c = a + (n-1)(b-a)

c = a + n(b-a) + a - b

n(b-a) = b + c - 2a

n = (b + c - 2a)/(b - 1)

Sum = (n/2)(a + c)

= (1/2)(b + c - 2a)(a + c)/(b - 1)