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