Question
Two APs have the same common difference. If the first term of these APs be $3$ and $8$ respectively, find the difference between the sums of their first $50$ terms.
✓
Answer
Let a and a' be the first term of two APs respectively.
Then, $a = 3$ and $a' = 8$
Let d be the common difference of two APs.
Let $S_{50}$ and $S'_{50}$ denote the sum of their first 50 temms.
Now, $\text{S}'_{50}-\text{S}_{50}=\frac{50}{2}\big[2(8)+49\text{d}\big]-\frac{50}{2}\big[2(3)+49\text{d}\big]$
$=25\big[16+49\text{d}\big]-25\big[6+49\text{d}\big]$
$25\big[16+49\text{d}-6-49\text{d}\big]$
$=25\times10$
$=250$
Then, $a = 3$ and $a' = 8$
Let d be the common difference of two APs.
Let $S_{50}$ and $S'_{50}$ denote the sum of their first 50 temms.
Now, $\text{S}'_{50}-\text{S}_{50}=\frac{50}{2}\big[2(8)+49\text{d}\big]-\frac{50}{2}\big[2(3)+49\text{d}\big]$
$=25\big[16+49\text{d}\big]-25\big[6+49\text{d}\big]$
$25\big[16+49\text{d}-6-49\text{d}\big]$
$=25\times10$
$=250$
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