If 5+9+13+.... to n terms/7+9+11+... to n terms=5/4, Find the value of n.
Answers
Answer:
answer is in attachment
Step-by-step explanation:
Given :-
(5+9+13+.... to n terms)/(7+9+11+... to n terms)=5/4
To find:-
Find the value of n.?
Solution:-
Given that :
(5+9+13+...to n terms)/(7+9+11+...to n terms) =5/4
On taking Numerator in the LHS
5+9+13+ ... (n terms)
First term = (a) = 5
Common difference (d)
= 9-5 = 4
= 13-9 = 4
d = 4
Since the common difference is same throughout the series they are in the AP
We know that
The sum of first n terms in an AP
Sn = (n/2)[2a+(n-1)d]
On Substituting these values in the above formula
=> Sn = (n/2)[2(5)+(n-1)(4)]
=> Sn = (n/2)[10+4n-4]
=> Sn = (n/2)[4n+6]
=> Sn = (n/2)(2)(2n+3)
=>Sn = n(2n+3)-----------------(1)
On taking Denominator in the LHS
7+9+11+ ... (n terms)
First term = (a) = 7
Common difference (d)
= 9-7 = 2
= 11-9 = 2
d = 2
Since the common difference is same throughout the series they are in the AP
We know that
The sum of first n terms in an AP
Sn = (n/2)[2a+(n-1)d]
On Substituting these values in the above formula
=> Sn = (n/2)[2(7)+(n-1)(2)]
=> Sn = (n/2)[14+2n-2]
=> Sn = (n/2)[2n+12]
=> Sn = (n/2)(2)(n+6)
=>Sn = n(n+6)-----------------(2)
Now,
(5+9+13+.... to n terms)/(7+9+11+... to n terms)=5/4
From (1)&(2)
=> [n(2n+3)]/[(n(n+6)] = 5/4
=> (2n+3)/(n+6) = 5/4
On applying cross multiplication then
=> 4(2n+3) = 5(n+6)
=> 8n +12 = 5n+30
=>8n -5n = 30-12
=> 3n = 18
=> n = 18/3
=> n = 6
Therefore,n = 6
Answer:-
The value of n for the given problem is 6
Used formulae:-
- The sum of first n terms in an AP=
- Sn = (n/2)[2a+(n-1)d]
- n = number of term
- a = First term
- d= Common difference