Diego wonders how much money he could save over 25 years if he puts $150 a year into an account with 4% interest compounded annually (once per year). He calculates the following, but thinks he must have something wrong, since he ended up with a very small amount of money:

\( \displaystyle \text{total amount} =150 \, \frac{1-0.04^{25}}{0.96} = 156.25\)

What did Diego forget in his calculation? How much should his total amount be? Explain or show your reasoning.

Which one of these equations is equivalent to \(8 = \frac{3+2x}{4+x}\) for \(x \neq \text-4\)?

Is \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\) an identity? Explain or show your reasoning.

Is \(a^4 + b^4 = (a+b)(a^3-a^2b-ab^2+b^3)\) an identity? Explain or show your reasoning.