in a geometric series, the sum of term a2 and a3 is 12

Question

and that the sum of term a3 and a4 is 60... 1st term is what?

Answer 1

Formula for the n^th term in the geometric series a_n = a r^n-1

a₂ = ar^{2 - 1}

a₂ = ar

a₃ = ar^{3 - 1}

a₃ = ar²

a₄ = a r^{4 - 1}

a₄ = ar³

ar + ar² = 12 ---> (1)

ar² + ar³ = 60 ---> (2)

Solve the equations (1) and (2).

From (2), r(ar + ar² ) = 60

From (1), r(12) = 60

r = 60/12

r = 5

Substitute r value in equation (1).

5a + 25a = 12

30a = 12

a = 12/30

a = 2/5.

First term in the sequence a = 2/5.