Question

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
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Answer 1

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 2

Answer is 12:19

Hope it helped you!

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