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Sn=a(rn−1)r−1S_n = \frac{a(r^n - 1)}{r - 1}Sn=r−1a(rn−1)
Sn=a(1−rn)1−rS_n = \frac{a(1 - r^n)}{1 - r}Sn=1−ra(1−rn)
First term, common ratio, number of terms
To avoid negative numbers in the denominator
Sn=anS_n = anSn=an
Would divide by zero (denominator r−1=0r - 1 = 0r−1=0)
Multiply sum by rrr, then subtract original equation
Most terms cancel, leaving only first and last terms
Trial-and-error or using logarithms
Compound interest and exponential growth patterns
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