Question

For the A.P. $-3,-7,-11,....$, can we find directly $a_{30}-a_{20}$ without actually finding $a_{30}$ and $a_{20}$? Give reasons for your answer.

Answer 1

Hint:- Use the general term formul for $n^{th}$ term of an AP, $a_n=a+(n-1)d$ and relations between them.

Given:-

$\text{a=first term,}\left(Let\right)$
$a_n$ $\text{=nth term},\left(Let\right)$
$\text{d=common difference between all the terms}\left(Let\right)$

Sequence : $-3,-7,-11,.....$

Step 1: Finding the common difference from the given terms

$d=-7-(-3)=4=-11-(-7)$

$\therefore d=-4$

Step 2: Replacing the value of $30$ for $n$ in the general term to get $a_{30}$

$a_{30}=a+(30-1)d=a+(30-1)d=a+29d$

Step 3: Replacing the value of $20$ for $n$ in the general term to get $a_{20}$

$a_{20}=a+(20-1)d=a+19d$

Step 4: Finding the difference between $a_{30}-a_{20}$

$a_{30}-a_{20}=(a+29d)-(a+19d)=10d=10\times (-4)=-40$

Final Step: Hence, $a_{30}-a_{20}=-40$. So, we can directly find them as first term completely gets cancelled during subtraction operation between the two terms.