Question

Find the sun of the following series. Round to the nearest hundredth if necessary.

4+8+16+…+2048

Answer 1

Sum=8188

Explanation:

This is a geometric series with a first term of 4 and a common ratio of 2. The formula for the sum of a geometric series is:

Sn​=1−ra(1−rn)​

where a is the first term, r is the common ratio and n is the number of terms. In this case, we have:

S11​=1−24(1−211)​

Simplifying, we get:

S11​=−14(−2047)​

S11​=8188

Therefore, the sum of the series is 8188.

Answer 2

4092

Explanation:

We can see that this is a geometric sequence where the first term is 4 and the common ratio is 2. We can use the formula for the sum of a geometric sequence to find the sum of this series:

sum = a(1 - r^n) / (1 - r)

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

We need to find n, the number of terms. We can use the formula for the nth term of a geometric sequence:

a_n = a * r^(n-1)

We want to find the value of n when a_n = 2048:

2048 = 4 * 2^(n-1)

512 = 2^(n-1)

n-1 = log2(512) = 9

n = 10

So there are 10 terms in the series. Now we can use the formula for the sum of a geometric sequence:

sum = a(1 - r^n) / (1 - r)

sum = 4(1 - 2^10) / (1 - 2)

sum = 4(1 - 1024) / (-1)

sum = 4(1023)

sum = 4092

Rounding to the nearest hundredth, the sum is approximately 4092.00.