If Ratio of the sum of the first n terms of two AP's is (7n+1): (4n+27). ,. Then find the ratio of their 9th terms.
Answers & Comments
Bunti360
Please understand carefully What I am going to do, I am going to use Comparative method here,
Let the two A.p's be a , b,
The sum of First n term's of A.P a is (n/2)*(7n + 1), We know that In general Sum of n terms is, (n/2) *(2a + (n-1)d),
We can generally guess that, in the ratio, n/2 must have been cancelled, So I have kept n/2 There,
Now For the First A.P i.e a, (n/2)(2a + nd - d) = n/2(7n + 1), => ((2a-d)+(nd)) = 7n + 1 (Now Here The Comparative method comes !), => We can say that by Comparison, nd = 7n and 2a-d = 1, => d = 7,and, 2a - 7 = 1, => 2a = 8, => a = 4,
Now we have all the values to find 9th term of A.P a ,
Now, Let us also find The Values of a and d in A.P b, I am going to do what I did there, => n/2 (2a + nd - d) = n/2(4n + 27), [n/2 gets cancelled now], => (2a-d) + nd = 4n + 27,
Again, Using Comparison, We can say that nd = 4n and, 2a-d = 27, => d = 4, and 2a- 4 = 27, => a = 31/2,
Now, We know that , Nth term of an A.P is a + (n-1)d, So, 9th term of A.P a is 4 + 8*7 = 4+56 = 60,
Now, 9th Term of A.P b is a + (n-1)d, => 31/2 + 8*4 = 15.5 + 32 = 47.5 ,
Now ratio of Both 9th terms of A.P is, 60/47.5 = 600/475 = 120/95 = 24/19,
Therefore the Ratio's of 9th Term's of both the A.p's is 24/19,
Hope you understand, Have a Great day !, Thanking you, Bunti 360 !
Answers & Comments
Let the two A.p's be a , b,
The sum of First n term's of A.P a is (n/2)*(7n + 1),
We know that In general Sum of n terms is,
(n/2) *(2a + (n-1)d),
We can generally guess that, in the ratio, n/2 must have been cancelled, So I have kept n/2 There,
Now For the First A.P i.e a,
(n/2)(2a + nd - d) = n/2(7n + 1),
=> ((2a-d)+(nd)) = 7n + 1 (Now Here The Comparative method comes !),
=> We can say that by Comparison,
nd = 7n and 2a-d = 1,
=> d = 7,and,
2a - 7 = 1,
=> 2a = 8,
=> a = 4,
Now we have all the values to find 9th term of A.P a ,
Now, Let us also find The Values of a and d in A.P b,
I am going to do what I did there,
=> n/2 (2a + nd - d) = n/2(4n + 27), [n/2 gets cancelled now],
=> (2a-d) + nd = 4n + 27,
Again, Using Comparison, We can say that nd = 4n and, 2a-d = 27,
=> d = 4, and
2a- 4 = 27,
=> a = 31/2,
Now, We know that , Nth term of an A.P is a + (n-1)d,
So, 9th term of A.P a is 4 + 8*7 = 4+56 = 60,
Now, 9th Term of A.P b is a + (n-1)d,
=> 31/2 + 8*4 = 15.5 + 32 = 47.5 ,
Now ratio of Both 9th terms of A.P is,
60/47.5 = 600/475 = 120/95 = 24/19,
Therefore the Ratio's of 9th Term's of both the A.p's is 24/19,
Hope you understand, Have a Great day !,
Thanking you, Bunti 360 !