The sum of the first two terms and the sum to infinity of a geometry progression are 48/7 and 7 respectively. find the values of the common ratio r and the first term when r is positive.

r = ±1/√7

a₁ = 7 − √7

Step-by-step explanation:

The first term is a₁ and the second term is a₁ r.

a₁ + a₁ r = 48/7

The sum of an infinite geometric series is S = a₁ / (1 − r)

a₁ / (1 − r) = 7

Start by solving for a₁ in either equation.

a₁ = 7 (1 − r)

Substitute into the other equation:

7 (1 − r) + 7 (1 − r) r = 48/7

1 − r + (1 − r) r = 48/49

1 − r + r − r² = 48/49

1 − r² = 48/49

r² = 1/49

r = ±1/√7

When r is positive, the first term is:

a₁ = 7 (1 − r)

a₁ = 7 (1 − 1/√7)

a₁ = 7 − 7/√7

a₁ = 7 − √7