# Lesson 26

Using the Sum

### 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?

### Solution

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### Problem 2

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

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

### Solution

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### 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.

### Solution

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### Problem 4

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

A:

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

B:

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

C:

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

D:

$$\frac{4+x}{8} = 3 + 2x$$

### Solution

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(From Unit 2, Lesson 23.)

### Problem 5

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

### Solution

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(From Unit 2, Lesson 24.)

### Problem 6

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

### Solution

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(From Unit 2, Lesson 24.)

### 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?

### Solution

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(From Unit 2, Lesson 25.)