If the sum of the first n terms of an arithmetic sequence is 165, the first term is -10
and the fourth term is -1, what is n?
Answers
Step-by-step explanation:
Given:-
the sum of the first n terms of an arithmetic sequence is 165, the first term is -10 and the fourth term is -1.
To find:-
The value of n?
Solution:-
First term of the given AP=(a)=-10
Fourth term of the AP=-1
we know that nth term=tn=a+(n-1)d
=>t4=a+(4-1)d
=>t4=a+3d
=>-10+3d=-1
=>3d=-1+10
=>3d=9
=>d=9/3
=>d=3
Common difference=3
And we know that the sum of first n terms in an AP=Sn=(n/2)[2a+(n-1)d]
Given the sum of first n terms=165
=>(n/2)[2(-10)+(n-1)(3)]=165
=>(n/2)[-20+(3n-3)]=165
=>(n/2)[3n-23]=165
=>(n)(3n-23)=165×2
=>n(3n-23)=330
=>3n²-23n=330
=>3n²-23n-330=0
on dividing by 3 both sides
=>(3n²/3)-(23n/3)-(330/3)=0
=>n²-(23n/3)-110=0
=>n²-(23n/3)=110
=>n²-2(n)(23/3)/2=110
=>n²-2(n)(23/6)=110
on adding (23/6)² both sides
=>n²-2(n)(23/6)+(23/6)²=110+(23/6)²
=>(n-(23/6))²=110+(529/36)
=>(n-(23/6))²=(3960+529)/36
=>(n-(23/6))²=4489/36
=>(n-(23/6)=±√(4489/36)
=>(n-(23/6)=±67/6
=>n=(23/6)±(67/6)
=>n=(23/6)+(67/6) or (23/6)-(67/6)
=>n=(23+67)/6 or (23-67)/6
=>n=90/6 or -44/6
=>n=15 or -22/3
Since n is the number of sides it can not be negative.
Therefore n=15
Answer:-
The value of n of the given AP=15
Used formulae:-
If a is the first term , d is the common difference of an AP then
- General term= tn=a+(n-1)d
- Sum of n terms=Sn=(n/2)[2a+(n-1)d]
- Completing the square method for solving n.