Question

Find the sum of the first 10 terms of the following geometric sequences:

{3, 6, 12, 24, 48...}


3066


3075


3069


3072

Answer 1

Answer: The sum of the first 10 terms of the given geometric sequence is 3069.

Explanation:

A geometric sequence is a sequence of numbers such that any two consecutive terms are in a constant ratio.

The first term of the given sequence is 3 and the common ratio is 2 (6/3 = 12/6 = 24/12 = ...).

To find the sum of the first 10 terms of a geometric sequence, we can use the formula:

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

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

So for this geometric sequence:

S = 3(1 - 2^10)/(1 - 2) = 3(1 - 1024)/(-1) = 3(-1023)/(-1) = 3069

Explanation: By using the formula for the sum of a geometric sequence, the sum of the first 10 terms of the sequence was found by substituting the first term, common ratio and number of terms into the formula.

Answer 2

C) 3069

Explanation:

A geometric series is the sum of the terms of a geometric sequence.

`\boxed{\begin{minipage}{7 cm}\underline{Sum of the first $n$ terms of a geometric series}\\\\$S_n=(a(1-r^n))/(1-r)$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $r$ is the common ratio.\\\end{minipage}}`

Given geometric sequence:

• {3, 6, 12, 24, 48...}

From inspection of the sequence, the first term is 3:

`\implies a=3`

To find the common ratio, divide consecutive terms:

`\implies r=(a_2)/(a_1)=(6)/(3)=2`

To find the sum of the first 10 terms, substitute the found values of a and r together with n=10 into the geometric series formula:

`\implies S_(10)=(3(1-2^(10)))/(1-2)`

`\implies S_(10)=(3(1-1024))/(1-2)`

`\implies S_(10)=(3(-1023))/(-1)`

`\implies S_(10)=(-3069)/(-1)`

`\implies S_(10)=3069`

Therefore, the sum of the first 10 terms of the given geometric sequence is:

• 3069