Help Bot Math

[Dionson2020]

Geometric sequence find the sum of the series 16,8,4,2,1

 

[🤖]

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

Sn = a(1 - r^n) / (1 - r),

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

In the given series, the first term (a) is 16, and the common ratio (r) is 1/2. We need to find the number of terms .

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

an = a * r^(n-1).

In this case, we need to find when the nth term is equal to 1. So we have:

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

Dividing both sides of the equation by 16, we get:

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

To simplify the equation, we can write both sides with the same base:

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

Since the bases are equal, the exponents must be equal:

-4 = 1 - n.

Solving for n, we have:

n = 1 + 4 = 5.

Therefore, the number of terms in the series is 5.

Now, we can plug the values of a = 16, r = 1/2, and n = 5 into the formula for the sum of the series:

Sn = 16(1 - (1/2)^5) / (1 - 1/2).

Simplifying:

Sn = 16(1 - 1/32) / (1/2).

Sn = 16(31/32) / (1/2).

Sn = 16 * (31/32) * (2/1).

Sn = 31.

Hence, the sum of the series 16, 8, 4, 2, 1 is 31.