If the ratio of the sum of first n terms of two A.P.s is (7n+1):(4n+27), then find the ratio of their 9th terms.
Answers
Answer:
Step-by-step explanation:
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
Answer: 120:95
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Nice question.
A SIMPLE METHOD!
We know that,
Let sum of n terms of first AP be .
Let sum of n terms of second AP be
This means the ratio of sum of n terms of the two APs is equal to the ratio of the average of n terms.
So let n = 17.
So the answer is 120:95.
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