Posts Tagged ‘infinite series’

Problem 755

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 = \frac{a_{1}}{1-r}, 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 = \frac{31,000,000(0.65)}{31,000,000}=0.65

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 = \frac{a_{1}}{1-r}

= \frac{31,000,000}{1-0.65}

Simplify

S = \frac{31,000,000}{1-0.65}

=\frac{31,000,000}{0.35}

\approx 88,571,429

Therefore , the total effect on the economy will be approximately 88,571,429.