Question

Q.19. If the ratio of the sum of first n
terms of two AP's is (7n + 1): (4n +27), find the ratio of mth terms .​

Answer 1

≡QUESTION≡

If the ratio of the sum of first n  terms of two AP's is (7n + 1): (4n +27), find the ratio of mth terms .​

                                                                 

║⊕ANSWER⊕║

GIVEN

Ratio of sum of n terms of two AP’s = (7n+1):(4n+27)

Consider the ratio these two AP’s m th terms as am : a’m →(2)

Recall the nth term of AP formula, an = a + (n – 1)d

Hence equation (2) becomes,

am : a’m = a + (m – 1)d : a’ + (m – 1)d’

On multiplying by 2, we get

am : a’m = [2a + 2(m – 1)d] : [2a’ + 2(m – 1)d’]

= [2a + {(2m – 1) – 1}d] : [2a’ + {(2m – 1) – 1}d’]

= S2m – 1 : S’2m – 1 

= [7(2m – 1) + 1] : [4(2m – 1) +27] [from (1)]

= [14m – 7 +1] : [8m – 4 + 27]

= [14m – 6] : [8m + 23]

Thus the ratio of mth terms of two AP’s is [14m – 6] : [8m + 23].

                                                       

Answer 2

24:19 is the answer


Given ratio of sum of n terms of two AP’s = (7n+1):(4n+27)
We can consider the 9th term as the m th term.
Let’s consider the ratio these two AP’s m th terms as am : a’m →(2)
Recall the nth term of AP formula, an = a + (n – 1)d
Hence equation (2) becomes,
am : a’m = a + (m – 1)d : a’ + (m – 1)d’
On multiplying by 2, we get
am : a’m = [2a + 2(m – 1)d] : [2a’ + 2(m – 1)d’]
= [2a + {(2m – 1) – 1}d] : [2a’ + {(2m – 1) – 1}d’]
= S2m – 1 : S’2m – 1 
= [7(2m – 1) + 1] : [4(2m – 1) +27] [from (1)]
= [14m – 7 +1] : [8m – 4 + 27]
= [14m – 6] : [8m + 23]
Thus the ratio of mth terms of two AP’s is [14m – 6] : [8m + 23].
now substitute the value of m as 9
so the answer becomes
120/95