A geometric series has first term a and common ratio r, where |r| < 1. The sum to infinity of the series is S. The sum to infinity of the series obtained by adding all the odd-number terms (i.e. first term + third term + fifth term + ...) is 4S/3. Find the value of r.
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Answer:
Sum to infinity = a / (1-r)
Sum to infinity of odd-number terms = a / (1-r²) => odd numbered terms are a, ar², ar4, ...
Sum to infinity / Sum to infinity of odd-number terms
= {a / (1-r)} / {a / (1-r²)}
= (1-r²) / (1-r)
= (1+r)(1-r) / (1-r)
= 1+r
But Sum to infinity / Sum to infinity of odd-number terms
= S / {4S/3}
= 3/4
Hence, 1+r = 3/4
r = -0.25
