Answer to Question #74619 in Calculus for Taryn

Question

If you save three pennies on January 1, six pennies on January 2, nine pennies on January 3, and continue this pattern for one year (not a leap year), what will be the value of your entire savings, in dollars, at the end of that one year? Express your answer as a decimal.

Answer 1

On the 1-st day I save 3=3⋅1 pennies, on the 2-nd day I save 6=3⋅2 pennies, on the 3-rd day I save 9=3⋅3pennies. So on n-th day i save 3npennies.
There are 365 days in any year that is not a leap year. So our target summ is :
3+6+9+...+3⋅365=3(1+2+3+4+ … +365)
Let’s add to (1+2+3+4+...+365) the same summ but taken in the other order:
(1+2+3+4+ … +365) + (365+364+ … +3+2+1) =
=(1+365)+(2+364)+(3+363)+ … +(364+2)+(365+1)=366⋅365
In the first equality we grouped summands in pairs first summand from the first pair of brackets with first from the second pair, second from the first pair of brackets with second from the second pair and so on.
In the second equality we used the fact, that summ inside each pair of brackets is equal to 366 and we have exactly 365 such pairs.
So (1+2+3+4+ … +365)=366⋅365/2 = 66795.
Now we saved 3+6+9+...+3⋅365=66795⋅3=200385 pennies or 2003.85 dollars