To solve for the sequence defined by [tex]\( a_n = (-1)^{n+1} n^2 \)[/tex], let's calculate the first few terms step by step.
Step-by-Step Calculation:
1. Calculate the first term [tex]\( a_1 \)[/tex]:
[tex]\[
a_1 = (-1)^{1+1} \cdot 1^2 = (-1)^2 \cdot 1 = 1 \cdot 1 = 1
\][/tex]
So, [tex]\( a_1 = 1 \)[/tex].
2. Calculate the second term [tex]\( a_2 \)[/tex]:
[tex]\[
a_2 = (-1)^{2+1} \cdot 2^2 = (-1)^3 \cdot 4 = -1 \cdot 4 = -4
\][/tex]
So, [tex]\( a_2 = -4 \)[/tex].
3. Calculate the third term [tex]\( a_3 \)[/tex]:
[tex]\[
a_3 = (-1)^{3+1} \cdot 3^2 = (-1)^4 \cdot 9 = 1 \cdot 9 = 9
\][/tex]
So, [tex]\( a_3 = 9 \)[/tex].
4. Calculate the fourth term [tex]\( a_4 \)[/tex]:
[tex]\[
a_4 = (-1)^{4+1} \cdot 4^2 = (-1)^5 \cdot 16 = -1 \cdot 16 = -16
\][/tex]
So, [tex]\( a_4 = -16 \)[/tex].
5. Calculate the fifth term [tex]\( a_5 \)[/tex]:
[tex]\[
a_5 = (-1)^{5+1} \cdot 5^2 = (-1)^6 \cdot 25 = 1 \cdot 25 = 25
\][/tex]
So, [tex]\( a_5 = 25 \)[/tex].
Summary of Results:
- [tex]\( a_1 = 1 \)[/tex]
- [tex]\( a_2 = -4 \)[/tex]
- [tex]\( a_3 = 9 \)[/tex]
- [tex]\( a_4 = -16 \)[/tex]
- [tex]\( a_5 = 25 \)[/tex]
Thus, the first five terms of the sequence [tex]\( a_n = (-1)^{n+1} n^2 \)[/tex] are [tex]\([1, -4, 9, -16, 25]\)[/tex].
