Step-by-step explanation:
The given sequence:
[tex]\dfrac{1}{2}+1+\dfrac{3}{2}+2+ ...[/tex]
Here, first term (a) = [tex]\dfrac{1}{2}[/tex], common difference(d) =[tex]1-\dfrac{1}{2}=\dfrac{1}{2}[/tex] and
the number of terms (n) = 25
The given sequence are in AP.
To find, the value of [tex]S_{25}[/tex] = ?
We know that,
The sum of nth terms of an AP
[tex]S_{n}=\dfrac{n}{2}[2a+(n-1)d][/tex]
The sum of 25th terms of an AP
[tex]S_{25}=\dfrac{25}{2}[2(\dfrac{1}{2})+(25-1)(\dfrac{1}{2})][/tex]
⇒ [tex]S_{25}=\dfrac{25}{2}[1+(24)(\dfrac{1}{2})][/tex]
⇒ [tex]S_{25}=\dfrac{25}{2}[1+12][/tex]
⇒ [tex]S_{25}=\dfrac{25}{2}[13][/tex]
⇒ [tex]S_{25}=\dfrac{325}{2}[/tex]
∴ [tex]S_{25}=\dfrac{325}{2}[/tex]
