This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with compound interest, but other than that, the textbook gave no hints really and I'm really not sure about how to approach it. Any guidance hints or help would be truly greatly appreciated. Thanks in advance :) So anyway, here the problem goes:
Compound interest: The ant and the grasshopper
Consider twin sisters with two different retirement savings plan:
Plan 1: Cora begins a retirement account at age $20$. She starts with $\$2000$ and then saves $\$2000$ per year at $7\%$ interest compounded annually for $10$ years. Then she stops contributing to the account but keeps her savings invested at the same rate.
Plan 2: Sally doesn't save any money in her twenties. When she turns $30$ she starts with $\$2000$ and then saves $\$2000$ per year at $7\% $ interest compounded annually for $35$ years.
Which one has more at age $65$ ?
At the end of the $10$ years, the first one has $$2000 \times (1.07^{10} + 1.07^9 + 1.07^8 + ... + 1.07^2 + 1.07) = 29567.20$$
This then is multiplied by $$1.07^{35} = 10.68$$ to yield $$315,676.62$$
The other one gets $$2000 \times (1.07 + 1.07^2 + 1.07^3 + ... + 1.07^{35}) = 2000 \times 137.4 = 274,800$$
She doesn't catch up.