今天做ASM study manual FM 的題目如下：
Jason deposits 3960 into a bank account at t = 0. The bank credits interest
at the end of each year at a force of interest (delta) = 1/(8+t). Interest
can be reinvested at an annual effective rate of 7%.
The total accumulated amount at time t = 3 is equal to X. Caculate X.
(A) 5394 (B) 5465 (C) 5551 (D) 5600 (E) 5685
答案是(A)，解答如下：
a(t) = (8+t)/8
i[t] = [a(t)/a(t-1)] -1 = 1/(7+t)
Interest earned each year:
3960*i[1] = 3960*(1/8) = 495
3960*i[2] = 3960*(1/9) = 440
3960*i[3] = 3960*(1/10) = 396
AV = 3960 + 495(1.07)^2 + 440(1.07) + 396 = 5393.53
原本的想法是覺得因為
a(t) = (8+t)/8 = 1 + t/8
所以就把他當成是 simple interest 的給息方式 ( i = 1/8 )
因此每年的 interest earned 都是
3960*(1/8) = 495
AV = 3960 + 495*(s3 angle 0.07) = 5551.37
剛好答案(C)也有這個數字...
想請問一下，這樣的想法是不是哪邊有錯誤?
若是往後又遇到 a(t) 的形式是 simple interest 的樣子，
能夠把他當成 simple interest 下去做嗎? 謝謝!