Hi! My teacher gave me a study guide a study guide, and there were a few sequence problems that I don't understand. Can someone please help?

Question

1. Write out the first three terms and the final term. Then find the sum. :

12

∑ 4^k

k=1

2. Given 5 − 15 + 45 − ⋯ − 645700815

a. Is this list sequence arithmetic or geometric? How can you tell?

b. Is Gauss or Euclid’s method preferable? Why?

c. Apply your chosen method to find the sum.

3. Find the first three terms and then the closed form for the following indefinite sum. :

n

∑ (6k-5)

k=0

4. Provide a closed form for the following indefinite sum:

1/4 + 1/16 + 1/64 + ... + 1/4ⁿ

Thank you!

Answer 1

This question requires a whole hour tutoring session, not just a few hints and/or answers.

A finite geometric series has sum (a-arⁿ⁺¹)/(1-r) ... but be careful about the first term is written.

The sum of the first n terms of the series 1+2+...+n is n(n+1)/2.

If r<1 in the infinite geometric series the sum is a/(1-r).