# Lesson 26

Using the Sum

- Let’s calculate some totals.

### Problem 1

Kiran plans to save $200 per year. Bank A would pay 6% interest, and Bank B would pay 4% interest (both compounded annually). How many years will it take to save $10,000 if he uses Bank A? Bank B?

### Problem 2

Find the sum of the first 20 terms of each sequence:

- \(1, \frac23, \frac49,\frac8{27},\frac{16}{81},\dots\)
- \(3, \frac63,\frac{12}9,\frac{24}{27},\frac{48}{81},\dots\)
- \(4,2,1,\frac12,\frac14,\dots\)

### Problem 3

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

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

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

### Problem 4

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

\(8 \cdot (4+x) = 3 + 2x\)

\(8 \cdot (3+2x) = 4 + x\)

\(8 - (4 + x) = 3 + 2x\)

\(\frac{4+x}{8} = 3 + 2x\)

### Problem 5

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

### Problem 6

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

### Problem 7

The formula for the sum \(s\) of the first \(n\) terms in a geometric sequence is given by \(s = a \left( \frac{1-r^{n}}{1-r}\right)\), where \(a\) is the initial value and \(r\) is the common ratio.

A medicine is prescribed for a patient to take 700 mg every 12 hours for 5 days. After 12 hours, 4% of the medicine is still in the body. How much of the medicine is in the body after the last dose?