Write a formula for the general term the nth term of each

Question

1. Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n find a₇, the seventh term of the sequence.

 

1. 1.5, -3, 6, -12, ....

 

2. 5, -1, 1/5, -1/25, ....

 

2. Use the formula for the sum of the first n terms of a geometric sequence to solve -

 

i. find the sum of the first 11 terms of the geometric sequence:

 

3, -6, 12, -24, ....

 

ii. find the sum of the first 11 terms of the geometric sequence:

 

4, -12, 36, -108, ....

 

3. A statement S_n about the positive integers is given. Write statements S_k and S_k+1, simplifying statement S_k+1 completely.

 

1. S_n: 2+7+12+...+(5n-3) = n(5n-1)/2.

 

2. S_n: 2 is a factor of n² - n +2.

 

4. Use mathematical induction to prove that each statement is true for every positive integer n.

 

1. 3+7+11+ ...+ (4n-1) = n(2n+1)

 

2. 2+7+12+...+(5n-3) = n(5n-1)/2.