If a geometric series has first term a and ratio r with |r|<1, what is the sum to infinity?

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Multiple Choice

If a geometric series has first term a and ratio r with |r|<1, what is the sum to infinity?

Explanation:
Sum to infinity for a geometric series with first term a and ratio r (with |r| < 1) is found by writing the series as S = a + ar + ar^2 + ar^3 + ... and then multiplying by r to get rS = ar + ar^2 + ar^3 + ... . Subtracting gives S − rS = a, so S(1 − r) = a, which yields S = a/(1 − r). The condition |r| < 1 ensures the tail shrinks and the sum converges to this finite value. This matches the form a/(1 − r). The other expressions don’t align with how the series is built: one would omit the initial term or assume a different sign or structure, so they don’t represent the sum to infinity.

Sum to infinity for a geometric series with first term a and ratio r (with |r| < 1) is found by writing the series as S = a + ar + ar^2 + ar^3 + ... and then multiplying by r to get rS = ar + ar^2 + ar^3 + ... . Subtracting gives S − rS = a, so S(1 − r) = a, which yields S = a/(1 − r). The condition |r| < 1 ensures the tail shrinks and the sum converges to this finite value. This matches the form a/(1 − r). The other expressions don’t align with how the series is built: one would omit the initial term or assume a different sign or structure, so they don’t represent the sum to infinity.

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