To solve the summation [tex]\(\sum_{n=2}^{10} 25(0.3)^{n+1}\)[/tex], we need to calculate the sum of the terms from [tex]\(n=2\)[/tex] to [tex]\(n=10\)[/tex], where each term is [tex]\(25 \times (0.3)^{n + 1}\)[/tex].
Let's break down the steps:
1. Define the general term: The general term of the summation is [tex]\(25 \times (0.3)^{n+1}\)[/tex].
2. Evaluate the terms: We need to evaluate this expression for each [tex]\(n\)[/tex] from 2 to 10.
- For [tex]\(n = 2\)[/tex]:
[tex]\[
25 \times (0.3)^{2 + 1} = 25 \times (0.3)^3 = 25 \times 0.027 = 0.675
\][/tex]
- For [tex]\(n = 3\)[/tex]:
[tex]\[
25 \times (0.3)^{3 + 1} = 25 \times (0.3)^4 = 25 \times 0.0081 = 0.2025
\][/tex]
- For [tex]\(n = 4\)[/tex]:
[tex]\[
25 \times (0.3)^{4 + 1} = 25 \times (0.3)^5 = 25 \times 0.00243 = 0.06075
\][/tex]
- For [tex]\(n = 5\)[/tex]:
[tex]\[
25 \times (0.3)^{5 + 1} = 25 \times (0.3)^6 = 25 \times 0.000729 = 0.018225
\][/tex]
- For [tex]\(n = 6\)[/tex]:
[tex]\[
25 \times (0.3)^{6 + 1} = 25 \times (0.3)^7 = 25 \times 0.0002187 = 0.0054675
\][/tex]
- For [tex]\(n = 7\)[/tex]:
[tex]\[
25 \times (0.3)^{7 + 1} = 25 \times (0.3)^8 = 25 \times 0.00006561 = 0.00164025
\][/tex]
- For [tex]\(n = 8\)[/tex]:
[tex]\[
25 \times (0.3)^{8 + 1} = 25 \times (0.3)^9 = 25 \times 0.000019683 = 0.000492075
\][/tex]
- For [tex]\(n = 9\)[/tex]:
[tex]\[
25 \times (0.3)^{9 + 1} = 25 \times (0.3)^{10} = 25 \times 0.0000059049 = 0.0001476225
\][/tex]
- For [tex]\(n = 10\)[/tex]:
[tex]\[
25 \times (0.3)^{10 + 1} = 25 \times (0.3)^{11} = 25 \times 0.00000177147 = 0.00004428675
\][/tex]
3. Sum the evaluated terms:
[tex]\[
0.675 + 0.2025 + 0.06075 + 0.018225 + 0.0054675 + 0.00164025 + 0.000492075 + 0.0001476225 + 0.00004428675 = 0.96426673425
\][/tex]
4. Compare with given options:
The sum we calculated is approximately [tex]\(0.96426673425\)[/tex].
Out of the given options:
- [tex]\(0.005\)[/tex]
- [tex]\(0.321\)[/tex]
- [tex]\(0.964\)[/tex]
- [tex]\(10.714\)[/tex]
The closest match to [tex]\(0.96426673425\)[/tex] is [tex]\(0.964\)[/tex].
Therefore, the answer is [tex]\(0.964\)[/tex].
