Answer :

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

Step-by-step 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.

semsee45

Answer:

C)  3069

Step-by-step explanation:

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

[tex]\boxed{\begin{minipage}{7 cm}\underline{Sum of the first $n$ terms of a geometric series}\\\\$S_n=\dfrac{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}}[/tex]

Given geometric sequence:

  • {3, 6, 12, 24, 48...}

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

[tex]\implies a=3[/tex]

To find the common ratio, divide consecutive terms:

[tex]\implies r=\dfrac{a_2}{a_1}=\dfrac{6}{3}=2[/tex]

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:

[tex]\implies S_{10}=\dfrac{3(1-2^{10})}{1-2}[/tex]

[tex]\implies S_{10}=\dfrac{3(1-1024)}{1-2}[/tex]

[tex]\implies S_{10}=\dfrac{3(-1023)}{-1}[/tex]

[tex]\implies S_{10}=\dfrac{-3069}{-1}[/tex]

[tex]\implies S_{10}=3069[/tex]

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

  • 3069

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