Question

The 3rd term of an AP is 3 and the 11th term is -21, find its first term and common difference.

Answer 1

✪AnSwEr

${\tt{\red{\underline{\large{Given}}}}}$

• 3rd term of an ap is 3
• 11th term =-21

${\sf{\green{\underline{\large{To\:find}}}}}$

• First term
• and
• common difference

${\sf{\pink{\underline{\Large{Explanation}}}}}$

Let the common difference and first term of an ap be a and d

Then

Tn = a+(n-1)d

$\tt{T_3=a+(3-1)d=3}$

$\rightarrow\tt{T_3=a+(3-1)d=3}$

$\rightarrow\tt{T_3=a+(2)d=3}$

$\rightarrow\tt{a+(2)d=3}$

$\rightarrow\tt{T_3=a+2d=3}$_____(1)

Now 11th term

$\tt{T_{11}=a+(11-1)d=-21}$

$\rightarrow\tt{T_={21}=a+(11-1)d=-21}$

$\rightarrow\tt{T_{21}=a+(10)d=-21}$

$\rightarrow\tt{a+10d=-21}$_____(2)

Now solving equations (1) and (2)

a+10d=-21

a+2d=3

=>8d=-24

=>d=-3

¶utting the value in Equation 2

a+2d=3

=>a-6=3

=>a=9

Hence,value of a is 9 and d=-3

Answer 2

$\sf\large\underline{Let:}$

• $\rm{The\: first\: term\:AP=a}$
• $\rm{Common\: difference=d}$

$\sf\large\underline{To\: Find:}$

• $\rm{1st\: term\:and\: Common\: difference\:AP=?}$

$\sf\large\underline{Solution:}$

• By applying formula of Tn to calculate first term and common difference ]

$\sf\large\underline{Formula\: used:}$

$\rm{\implies T_{n}=a+(n-1)d}$

$\sf\underline{Given\:in\:1st\:case:}$

$\rm{\implies The\:3rd\:term\:of\:AP=3}$

$\rm{\implies T_{3}=a+(3-1)d}$

$\rm{\implies a+2d=3----(i)}$

$\sf\underline{Given\:in\:2nd\:case:}$

$\rm{\implies The\:11th\:term\:of\:AP=-21}$

$\rm{\implies T_{11}=a+(11-1)d}$

$\rm{\implies a+10d=-21-----(ii)}$

• Now, solving equation 1 and 2 ]

$\rm{\implies a+2d=3}$

$\rm{\implies a+10d=-21}$

• by solving equation we get here]

$\rm{\implies -8d=24}$

$\rm{\implies d=-3}$

• putting the value of d=-3 in eq 1]

$\rm{\implies a+2d=3}$

$\rm{\implies a+2(-3)=3}$

$\rm{\implies a-6=3}$

$\rm{\implies a=3+6}$

$\rm{\implies a=9}$

$\sf\large{Hence,}$

$\tt{\implies Common\:difference=-3}$

$\tt{\implies First\:term=9}$