It is a well-known fact the sum of counting numbers from 1 to 100 adds up to=5050. Now, suppose that these numbers are dollars deposited into an investment account at a rate of 5% compounded annually, in this manner:
Year 1=$1, year 2=$2, year 3=$3.......and so on till 100th year=$100. What would the total sum be in 100 years. P.S. My teacher says there is a short cut to this calculation!. Any help would be great. Thanks.
+130560 Assuming that the deposits are made at the beginning of each year ....we have....
1(1.05)^100 + 2(1.05)^99 + 3(1.05)^98 + ...... + 99(1.05)^2 + 100(1.05)
n = 99
∑ (n + 1)(1.05)^(100-n) = about $55,451.05
n = 0
Maybe heureka knows a shortcut ???
+130560 OK....assuming end of year deposits, we have
1(1.05)^99 + 2(1.05)^98 + 3(1.05)^97 + ....+ 99(1.05)^1 + 100(1.05)^0
n = 100
∑ (n)(1.05)^(100-n) ≈ $52,810.53
n = 1
Very good effort CPhill, but the answer in my book is different by a few thousands dollars. This might be up in "heureka's alley!". I shall get the "shortcut" from our teacher tomorrow.
+130560 OK...sorry....I did my best......if heureka doesn't provide us with an answer, why don't you post the shortcut on here....????......
Thanks, CPhill
Sorry, CPhill: After my 4th. comment above, I went to bed! It turns out that the answer is surprisingly simple. It is a relatively common and well-known TVM formula widely used in advanced accounting. And it is this:
FV= G {[1 + R]^N - RN - 1} / R^2=
FV= G {[ 1 + .05]^100 -(.05*100) - 1} / .05^2. Where:
G is a steadily increasing payment amount, that starts at G and increases by G for each subsequent period.
FV= 1{ 131.50125... - 5 - 1} / 0.0025
FV= {125.50125} / .0025
FV=$50,200.50.
And that is the answer in the book. Great effort on your part though, and I thank you.
