Question

The sum of an infinite geometric series whose first term is a and common ratio is r is given by

Answer 1

Answer:

a / ( 1 - r )

[ Note that this only holds if |r| < 1, as otherwise the terms in the sequence just get bigger and bigger so the sum of the series doesn't tend to any value. ]

Step-by-step explanation:

a + ar + ar² + ar³ + ...

= a ( 1 + r + r² + r³ + ... )

= a / ( 1 - r )

If the last step seems mysterious, read on...

S = 1 + r + r² + r³ + ...

rS = r + r² + r³ + ...

Subtracting the second of these from the first one:

S - rS = 1

=> S = 1 / ( 1 - r )