Write the first five terms of the sequence using the given rule. In each case, the initial value of the index is 1.
` a_(n) = n^(2) +5 `

Answer

To find the first five terms of the sequence defined by the rule \( a_n = n^2 + 5 \) with the initial value of the index as 1, we will substitute values of \( n \) from 1 to 5 into the formula.

**Step 1:** Calculate \( a_1 \)
- Substitute \( n = 1 \) into the formula:
\[ a_1 = 1^2 + 5 = 1 + 5 = 6 \]

**Step 2:** Calculate \( a_2 \)
- Substitute \( n = 2 \) into the formula:
\[ a_2 = 2^2 + 5 = 4 + 5 = 9 \]

**Step 3:** Calculate \( a_3 \)
- Substitute \( n = 3 \) into the formula:
\[ a_3 = 3^2 + 5 = 9 + 5 = 14 \]

**Step 4:** Calculate \( a_4 \)
- Substitute \( n = 4 \) into the formula:
\[ a_4 = 4^2 + 5 = 16 + 5 = 21 \]

**Step 5:** Calculate \( a_5 \)
- Substitute \( n = 5 \) into the formula:
\[ a_5 = 5^2 + 5 = 25 + 5 = 30 \]

Now that we have calculated all five terms, we can write them down:

The first five terms of the sequence are:
\[ 6, 9, 14, 21, 30 \]