if the ratio of the sum of first n terms of two ap is 7 and + 14 and + 27 find the ratio of their 9th term
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24 : 19 is the 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 nth term.
Let’s consider the ratio these two AP’s nth terms as an : a’n →(2)
Recall the nth term of AP formula, an = a + (n – 1)d
Hence equation (2) becomes,
an : a’n = a + (n – 1)d : a’ + (n – 1)d’
On multiplying by 2, we get
an : a’n = [2a + 2(n – 1)d] : [2a’ + 2(n – 1)d’]
= [2a + {(2n – 1) – 1}d] : [2a’ + {(2n – 1) – 1}d’]
= S2n – 1 : S’2n – 1
= [7(2n – 1) + 1] : [4(2n – 1) +27] [from (1)]
= [14n – 7 +1] : [8n – 4 + 27]
= [14n – 6] : [8n + 23]
Thus the ratio of nth terms of two AP’s is [14n – 6] : [8n + 23].
now substitute n as 9
therefore the answer becomes
120/95 = 24/19 = 24 : 19
hope it helps u
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