To evaluate the sum [tex]\(\sum_{n=2}^{10} 25(0.3)^{n+1}\)[/tex], we'll proceed step by step.
### Step 1: Understanding the Series Expression
The general term in the sum is [tex]\(25(0.3)^{n+1}\)[/tex]. So, we need to calculate this for each integer [tex]\(n\)[/tex] from 2 to 10, and then sum all these terms.
### Step 2: Compute Each Term individually for [tex]\(n\)[/tex] from 2 to 10
1. When [tex]\(n = 2\)[/tex]:
[tex]\[
25(0.3)^{2+1} = 25(0.3)^3 = 25 \times 0.027 = 0.675
\][/tex]
2. When [tex]\(n = 3\)[/tex]:
[tex]\[
25(0.3)^{3+1} = 25(0.3)^4 = 25 \times 0.0081 = 0.2025
\][/tex]
3. When [tex]\(n = 4\)[/tex]:
[tex]\[
25(0.3)^{4+1} = 25(0.3)^5 = 25 \times 0.00243 = 0.06075
\][/tex]
4. When [tex]\(n = 5\)[/tex]:
[tex]\[
25(0.3)^{5+1} = 25(0.3)^6 = 25 \times 0.000729 = 0.018225
\][/tex]
5. When [tex]\(n = 6\)[/tex]:
[tex]\[
25(0.3)^{6+1} = 25(0.3)^7 = 25 \times 0.0002187 = 0.0054675
\][/tex]
6. When [tex]\(n = 7\)[/tex]:
[tex]\[
25(0.3)^{7+1} = 25(0.3)^8 = 25 \times 0.00006561 = 0.00164025
\][/tex]
7. When [tex]\(n = 8\)[/tex]:
[tex]\[
25(0.3)^{8+1} = 25(0.3)^9 = 25 \times 0.000019683 = 0.000492075
\][/tex]
8. When [tex]\(n = 9\)[/tex]:
[tex]\[
25(0.3)^{9+1} = 25(0.3)^{10} = 25 \times 0.0000059049 = 0.0001476225
\][/tex]
9. When [tex]\(n = 10\)[/tex]:
[tex]\[
25(0.3)^{10+1} = 25(0.3)^{11} = 25 \times 0.00000177147 = 0.00004428675
\][/tex]
### Step 3: Sum All the Calculated Terms
Now, we sum all these individual terms:
[tex]\[
0.675 + 0.2025 + 0.06075 + 0.018225 + 0.0054675 + 0.00164025 + 0.000492075 + 0.0001476225 + 0.00004428675 = 0.96426673425
\][/tex]
### Step 4: Rounding (if needed)
The final sum, [tex]\(0.96426673425\)[/tex], is approximately equal to [tex]\(0.964\)[/tex] when rounded to three decimal places.
Thus, the value of the sum [tex]\(\sum_{n=2}^{10} 25(0.3)^{n+1}\)[/tex] is closest to:
[tex]\[
\boxed{0.964}
\][/tex]
