Find the sum given below: $7 + 10 \frac{1}{2} + 14 + \ldots + 84$

The given series is $7 + 10 \frac{1}{2} + 14 + \ldots + 84$.
This is an Arithmetic Progression $(A.P.)$ where:
First term $(a)$ = $7$
Last term $(l)$ = $84$
Common difference $(d)$ = $10 \frac{1}{2} - 7 = \frac{21}{2} - 7 = \frac{7}{2}$.
Let $84$ be the $n^{\text{th}}$ term of this $A.P.$
Using the formula $l = a + (n - 1)d$:
$84 = 7 + (n - 1) \frac{7}{2}$
$77 = (n - 1) \frac{7}{2}$
$11 = (n - 1) \frac{1}{2}$
$22 = n - 1$
$n = 23$.
Now,find the sum using the formula $S_n = \frac{n}{2}(a + l)$:
$S_{23} = \frac{23}{2}(7 + 84)$
$S_{23} = \frac{23 \times 91}{2}$
$S_{23} = \frac{2093}{2}$
$S_{23} = 1046 \frac{1}{2}$.