F5 math problems ( Summing a sequence )

2007-09-20 1:30 am

1.An insurance company promotes a ten-year life insurance with zero cost. In the scheme,$10000 is deposited by the insurer on the same insurance date of the year for ten years.After ten years, a total of $100000 is returned to the insurer.Assume that the interest rate is 4%p.a.,which is compounded annually.Find the actual cost to the nearest dollar of the life insurance policy.

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2.The sum to infinity of an infinite geometric sequence is 16 and the sum to infinity of the squares of its terms is 153.6 . Find the common ratio and the fourth term of the sequence.

回答 (1)

Oliver

2007-09-20 2:19 am

✔ 最佳答案

1.
a = first term
r = common ratio
GP sum = a [ (r^n) –1 ] / (r-1)

a = 10000
r = 1.04
GP sum = 10000 [ (1.04^10) – 1] / (1.04 – 1)
= 120061 (to the nearest dollar)

Therefore actual cost is 120061 – 100000
= $20061

2.
GP sum to infinity = a / (1 – r)
The sum to infinity of an infinite geometric sequence is 16 means:
a / (1 – r) = 16
a = 16(1-r)

The sum to infinity of the squares of its terms is 153.6 means:
a would become a^2
r would become r^2
a^2 / (1 – r^2) = 153.6
256 ( 1 – 2r + r^2 ) = 153.6 – 153.6 r^2
256 – 512r + 256r^2 = 153.6 – 153.6r^2
409.6r^2 – 512r + 102.4 = 0
4r^2 – 5r +1 = 0
r = 1/4 or 1 (rejected because r cannot be 1 or otherwise the 分母would equal to 0 )
a = 12
4th term is 12 x (1/4)^3 = 3/16

參考: 自己

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