5. Find the sum of the following series
(iv) $21 + 22 + 23 + . . . . . + 51$

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Solution

Step 1: Note the given data
Given series $21 + 22 + 23 + . . . . . + 51$
First-term $t_{1} = 21$
Second-term $t_{2} = 22$
Third-term $t_{3} = 23$
Step 2: Find the common difference
The general formula of common difference $d = {(n + 1)}^{t h} t e r m - n^{t h} t e r m$
Common difference $22 - 21$
$= 1$
Similarly, $d = t_{2} - t_{1} = t_{3} - t_{2} = 1$
Given series is an Arithmetic series
Step 3: Finding $n^{t h}$ term of A.P in term of $n$ .
The general formula of $n^{t h}$ term of A.P. is $t_{n} = t_{1} + (n - 1) d$
$= 21 + (n - 1) \times 1 = 21 + n - 1 = 20 + n$
Step 4: Comparing the $n^{t h}$ term with the given data
Let $t_{n} = 51$
Comparing and we get
$20 + n = 51$
$\Rightarrow n = 51 - 20 \Rightarrow n = 31$
Step 5: Find the sum of the given series
The general formula of the sum of the series $= \frac{n}{2} (t_{1} + t_{n})$
The sum is
$= \frac{31}{2} (21 + 51) = \frac{31}{2} \times 72 = 31 \times 36 = 1116$
Hence, the sum of the given series $21 + 22 + 23 + . . . . . + 51$ is $1116$ .

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