If the ratio of the sums of first n terms of two A.P.s is 5n+13 / 7n +27 then the ratio of their 4th terms is__________.plz ans my question
Answers
Step-by-step explanation:
Given :-
The ratio of the sums of first n terms of two A.P.s is 5n+13 / 7n +27 .
To find :-
Find the ratio of their 4th terms of the two APs?
Solution :-
Let the first term of the first AP be 'a'
Let the common difference of the first term be 'd'
Then,
The Sum of the first n terms of the AP = Sn
=> Sn = (n/2)[2a+(n-1)d] --------------(1)
Let the first term of the second AP be 'b'
Let the common difference of the second AP be 'c'
The sum of the first n terms of the second AP
=> Sn = (n/2)[2b+(n-1)c] -------------(2)
The ratio of the sum of the first n terms of two APs
=> (n/2)[2a+(n-1)d] : (n/2)[2b+(n-1)c]
=> (n/2)[2a+(n-1)d] / (n/2)[2b+(n-1)c]
=> [2a+(n-1)d] / [2b+(n-1)c] --------(3)
According to the given problem
The ratio of the sums of first n terms of two A.P.s = (5n+13) : (7n +27 )
= (5n+13) / (7n +27 ) ---------------(4)
(3) and (4) are equal.
=> [2a+(n-1)d]/[2b+(n-1)c] = (5n+13)/(7n+27)
To find ratio of 4 terms , write n = 7 since (2×4-1)=7
=>[2a+(7-1)d]/[2b+(7-1)c]=(5(7)+13))/(7(7)+27))
=> [2a + 6d] / [2b+6c] = (35+13) / (49+27 )
=> [2a + 6d] / [2b+6c] = 48 / 76
=> 2(a+3d) / 2(b+3c) = 12 / 19
=> (a+3d) / (b+3c) = 12/19
=> a4 / b4 = 12/19
Since nth term of an AP = an = a+(n-1)d
=> a4 : b4 = 12:19
a4 is the 4 th term of the first AP
b4 is the 4 th term of the second AP
Answer:-
The ratio of 4th terms of the two APs is 12:19
Used formulae:-
- nth term of an AP = an = a+(n-1)d
- The Sum of the first n terms of the AP = Sn
- => Sn = (n/2)[2a+(n-1)d]
- a = First term
- d = Common difference
- n =Number of terms
- Sn = Sum of the first n terms
- an = nth or General term in an AP.
Answer is 12:19
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