An arithmetic progression is a sequence of numbers in which adjacent numbers differ by a constant factor.
It is represented in the form
a,a+d,a+2d,... a is known as the first term of the A.P and
d the common difference.
Let "a" be the first term and "d" the common difference of the A.P. Then we have
a21+a22+a23=261 (i)
{ Because the sum of the last three terms is
261.} The three middle terms are
a11,a12 and
a13. So, we have
a11+a12+a13=141 (ii)
From equation (i) we have
a+(21−1)d+a+(22−1)d+a+(23−1)d=261 ⇒a+20d+a+21d+a+22d=261 ⇒3a+63d=261 (iii)
From equation (ii) we have
a+(11−1)d+a+(12−1)d+a+(13−1)d=141 ⇒a+10d+a+11d+a+12d=141 ⇒3a+33d=141 (iv)
Subtracting equation (iv) from equation (iii),
we get
3a+63d−(3a+33d)=261−141 ⇒3a+63d−3a−33d=120 ⇒30d=120 Dividing both sides by 30 , we get
‌=‌=4 i.e.
d=4.
Substituting the value of
d in equation (iv), we get
3a+33×4=141 ⇒3a+132=141 ⇒3a=9 ⇒a=3 Hence the first term of the A.P
=3.