An infinite geometric series has a sum of 20, where all the terms are positive. The sum of the first and second terms are 12.8. What is the first term?

1 Answer
Noah G
Mar 30, 2018

#a = 8#

Explanation:

Recall the sum of the terms in an infinite geometric series is #20#.

#S_n = a/(1 - r)#

We know the sum of the series, so:

#20 = a/(1 - r)#

#a = 20(1 - r)#

The first term is #a#. The second term is #ar#. Therefore, #s_2 = a + a r = a(1 + r)#. We know the sum of the first two terms is #12.8 = 64/5#. Therefore our second equation becomes #a(1 + r) = 64/5#.

Substituting the first equation into the second we get:

#(20(1 - r))(1 +r) = 64/5#

#(20 - 20r)(1 + r ) = 64/5#

#20 - 20r + 20r - 20r^2 = 64/5#

#100 - 100r^2 = 64#

#36 = 100r^2#

#0.36 = r^2#

#r = +- 0.6#

Since all the terms are positive, #r = +0.6# (if the common ratio was negative, every two terms would be negative).

Since #a = 20(1 - r)#, #a = 20(0.4) = 8#

Hopefully this helps!

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