Answer:
22.5
Step-by-step explanation:
If you expand the series, you can see the first few terms of the series:
- Putting 1 in [tex]n[/tex], [tex]\frac{1}{2}(1)-\frac{1}{2}=0[/tex]
- Putting 2 in [tex]n[/tex], [tex]\frac{1}{2}(2)-\frac{1}{2}=0.5[/tex]
- Putting 3 in [tex]n[/tex], [tex]\frac{1}{2}(3)-\frac{1}{2}=1[/tex]
- Putting 4 in [tex]n[/tex], [tex]\frac{1}{2}(4)-\frac{1}{2}=1.5[/tex]
We can see the series is 0, 0.5, 1, 1.5, ....
This is an arithmetic series with common difference (the difference in 2 terms) 0.5 and first term 0.
We know formula for sum of arithmetic series:
[tex]s_{n}=\frac{n}{2}(2a+(n-1)d)[/tex]
Where,
- [tex]S_{n}[/tex] denotes the nth partial sum
- [tex]a[/tex] is the first term (in our case it is 0)
- [tex]n[/tex] is the term (in our case it is 10 since we want to find 10th partial sum -- sum until first 10 terms)
- [tex]d[/tex] is the common difference (difference in term and the previous term) (in our case it is 0.5)
Substituting these into the formula, we get the 10th partial sum to be:
[tex]s_{10}=\frac{10}{2}(2(0)+(10-1)(0.5))\\s_{10}=5(0+(9)(0.5))\\s_{10}=5(0+4.5)\\s_{10}=5(4.5)\\s_{10}=22.5[/tex]
So the sum of the first 10 terms is 22.5. Third answer choice is right.
