in an ap the sum of first 11 terms is 44 and the sum of next 11 terms is 55 find ap
Answers
Step-by-step explanation:
Given:-
The sum of first 11 terms is 44 and the sum of next 11 terms is 55 .
To find:-
Find the AP
Solution:-
Given that
The sum of first 11 terms = 44
=>a1+a2+a3+...+a11 = 44
=>a,a+d,a+2d,...,a+10d = 44
We know that
the sum of first n terms of AP
=Sn=(n/2)[2a+(n-1)d]
=> S11 = 44
=> (11/2)[2a+(11-1)d] = 44
=> (11/2)[2a+10d ]=44
=> (11/2)×2[a+5d] = 44
=> 11(a+5d)=44
=>a+5d = 44/11
=>a+5d =4---------(1)
and given that
The sum of next 11 terms = 55
a12+a13+...+a22=55
=>(a+11d)+(a+12d)+----+(a+21d)=55
S(next 11 terms) =
(11/2)[2(a+11d)+(11-1)(d] = 55
=>(11/2)[2a+22d+10d]=55
=>(11/2)[2a+32d]=55
=>(11/2)×2(a+16d)=55
=>11(a+16d)=55
=>a+16d = 55/11
a+16d = 5----------(2)
On subtracting (1) from (2)
a+16d = 5
a+5d = 4
(-)
_________
0+11d = 1
________
=>11d = 1
=>d = 1/11
On Substituting the value of d in (1) then
a+5(1/11)=4
=>a+(5/11) = 4
=>a = 4-(5/11)
=>a = (44-5)/11
=>a = 39/11
First term = 39/11
common difference = 1/11
The general form of an AP = a,a+d,a+2d,...
a=39/11
a+d = (39/11)+(1/11)
=>(39+1)/11
=>40/11
a+d = 40/11
a+2d
(39/11)+2(1/11)
=>(39/11)+(2/11)
=>(39+2)/11
=>41/11
The AP: 39/11 , 40/11 , 41/11,...
Answer:-
The required AP is 39/11 , 40/11 , 41/11,...
Used formulae:-
- The general form of an AP = a,a+d,a+2d,...
- the sum of first n terms of AP
- =Sn=(n/2)[2a+(n-1)d]
- a= First term
- d=common difference
- n=number of terms