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The limiting sum of the GP is twice the sum of the first two terms. Find the possible values of the common ratio?

Mar 26

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Mar 26

Let the first term of the GP be a and the common ratio be r.

The sum of the first two terms of the GP is a + ar = a(1 + r).

The limiting sum of the GP is given by S = a/(1 - r) when |r| < 1.

According to the given condition, the limiting sum is twice the sum of the first two terms:

2(a + ar) = a/(1 - r)

2a + 2ar = a/(1 - r)

2a + 2ar = a(1 + r)

2 + 2r = 1 + r

r = -1

Therefore, the only possible value of the common ratio is -1.

The sum of the first two terms of the GP is a + ar = a(1 + r).

The limiting sum of the GP is given by S = a/(1 - r) when |r| < 1.

According to the given condition, the limiting sum is twice the sum of the first two terms:

2(a + ar) = a/(1 - r)

2a + 2ar = a/(1 - r)

2a + 2ar = a(1 + r)

2 + 2r = 1 + r

r = -1

Therefore, the only possible value of the common ratio is -1.