The government, through a subsidy program, distributes 31,000,000. If each person or agency spends 65% of what is received, and 65% of this is spend, and so on, how much total increase in spending results from this government action?
Solution:-
The total increase in spending is given by infinite series shown below.
31,000,000 + 31,000,000 (0.65) + 31,000,000(0.65)2 + …..
Thus , the series is geometric. When -1 < r < 1, the sum of an infinite geometric series is given by S∞ = 
, where r is the common ratio and a1 is the first term of the geometric series.
To find the common ratio, choose a term the sequence (other than the first) and divide by the preceding term.
r = 
Since -1 < 0.65 < 1 , the series has a sum.
Use the formula to find the sum of the infinite geometric series. Substitute 0.65 for r and 31,000,000 for a1.
S∞ =
= 
Simplify
S∞ =
=
88,571,429
Therefore , the total effect on the economy will be approximately 88,571,429.