1. Find the sum of the sequence: 32, -16, 8, -4, 2, ....

My answer is 64/3. Am I correct?

32/(1 + 1/2) = 64/3

you are correct

To find the sum of a sequence, we can use the formula for the sum of a finite geometric series:

S = a * (r^n - 1) / (r - 1)

- where S is the sum of the sequence, a is the first term, r is the common ratio, and n is the number of terms.

In this case, the first term (a) is 32, and the common ratio (r) is -1/2, because each term is half of the previous term. We need to find the number of terms (n).

To find the number of terms (n), we can use the formula for the nth term of a geometric sequence:

a_n = a * r^(n-1)

- where a_n is the nth term.

In this case, the nth term (a_n) is 2, and the first term is 32. Let's solve for n:

2 = 32 * (-1/2)^(n-1)

To simplify this equation, we can divide both sides by 32:

1/16 = (-1/2)^(n-1)

Since 1/16 is equal to (-1/2)^4, we can write:

(-1/2)^(n-1) = (-1/2)^4

By comparing the exponents, we can conclude that:

n-1 = 4

Therefore, n = 5.

Now that we know the number of terms (n), we can substitute the values into the sum formula:

S = 32 * ((-1/2)^5 - 1) / (-1/2 - 1)
S = 32 * (-(1/32) - 1) / (-3/2)
S = 32 * (-33/32) / (-3/2)
S = 33/2
S = 16.5

So, the correct sum of the sequence is 16.5. Your answer of 64/3 is not correct.