Question

The sums of the first n terms of two A.P.s are in the ratio (7n+2): (n+4). Find the ratio of their 5th terms.
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Answer 1

Answer:

Given that, S1S2=7n+14n+27,S1S2=7n+14n+27,

Sum of first ‘n’ terms of AP, whose first term is ‘a’ and common difference is ‘d’, is given by-

S=n2[2a+(n−1)d]S=n2[2a+(n−1)d]

Let,

First term and common difference of two AP’s having sum S1S1 and S2S2 be ‘a’ , ’d’ and ‘A’ , ‘D’ respectively.

Then, S1S2=n2[2a+(n−1)d]n2[2A+(n−1)D],S1S2=n2[2a+(n−1)d]n2[2A+(n−1)D],

=> S1S2=2a+(n−1)d2A+(n−1)DS1S2=2a+(n−1)d2A+(n−1)D

=> S1S2=a+n−12dA+n−12DS1S2=a+n−12dA+n−12D ————-(1)

Now,nth term of an AP having first term ‘a’ and common difference ‘d’ is given as,

Tn=a+(n−1)dTn=a+(n−1)d

Thus, 9th term of the two AP’s having sum S1S1 and S2S2 will be a+8da+8d (Let it be A1A1) and A+8DA+8D (Let it be A2A2) respectively,

A1A2=a+8dA+8DA1A2=a+8dA+8D ————-(2)

Comparing equations (1) and (2), we get-

n−12=8n−12=8 or n=17n=17

Given equation,a+n−12dA+n−12D=7n+14n+27a+n−12dA+n−12D=7n+14n+27

Putting n=17n=17 in the above equation,we will get-

a+17−12dA+17−12D=7(17)+14(17)+27a+17−12dA+17−12D=7(17)+14(17)+27

a+162dA+162D=7(17)+14(17)+27a+162dA+162D=7(17)+14(17)+27

a+8dA+8D=A1A2=7(17)+14(17)+27a+8dA+8D=A1A2=7(17)+14(17)+27

Thus, A1A2=12095=2419.

Step-by-step explanation:

Answer 2

Answer

• 5 : 1

Given

• The sums of the first n terms of two A.P.s are in the ratio (7n+2): (n+4)

To Find

• Ratio if their 5th terms

Solution

Let first term , common difference and sum of n terms of first AP be a₁ , d₁ , S₁ .

⇒ S₁ = ⁿ/₂ [ 2a₁ + ( n - 1 ) d₁ ] and

S₂ = ⁿ/₂ [ 2a₂ + ( n - 1 ) d₂ ]

A/c , " The sums of the first n terms of two A.P.s are in the ratio (7n+2): (n+4) "

⇒ S₁ / S₂ = ⁿ/₂ [ 2a₁ + ( n - 1 ) d₁ ] /  ⁿ/₂ [ 2a₂ + ( n - 1 ) d₂ ]

⇒ ( 7n + 2 ) / ( n + 4 ) = ( 2a₁ + ( n - 1 ) d₁ ) / ( 2a₂ + ( n - 1 ) d₂ )

Sub. 9 in place of n , we get ,

⇒ ( 7 (9) + 2 ) / ( 9 + 4 ) =  ( 2a₁ + ( 9 - 1 ) d₁ ) / ( 2a₂ + ( 9 - 1 ) d₂ )

⇒ 63+2 / 13 = (2a₁ + 8d) / (2a₂ + 8d₂)

⇒ 65 / 13 = ( a₁ + 4d₁ ) / ( a₂ + 4d₂ )

⇒ 5 / 1 = [ a₁ + (5-1) d₁ ] / [ a₂ + (5-1) d₂ ]

So , Ratio of their 5th terms is 5 : 1