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Nov 13, 2013

# The ratio of the sum of n terms of two A.P.s is (7n+1):(4n+27).

The ratio of the sum of n terms of two A.P.s is (7n+1):(4n+27).Find the ratio of their mth term.

Priyanshu

Member since Nov 29, 2016

Given ratio of sum of n terms of two APâ€™s = (7n+1):(4n+27) Letâ€™s consider the ratio these two APâ€™s mth terms as am : aâ€™m â†’(2) Recall the nth term of AP formula, an = a + (n â€“ 1)d Hence equation (2) becomes, am : aâ€™m = a + (m â€“ 1)d : aâ€™ + (m â€“ 1)dâ€™ On multiplying by 2, we get am : aâ€™m = [2a + 2(m â€“ 1)d] : [2aâ€™ + 2(m â€“ 1)dâ€™] = [2a + {(2m â€“ 1) â€“ 1}d] : [2aâ€™ + {(2m â€“ 1) â€“ 1}dâ€™] = S2m â€“ 1 : Sâ€™2m â€“ 1 = [7(2m â€“ 1) + 1] : [4(2m â€“ 1) +27] [from (1)] = [14m â€“ 7 +1] : [8m â€“ 4 + 27] = [14m â€“ 6] : [8m + 23] Thus the ratio of mth terms of two APâ€™s is [14m â€“ 6] : [8m + 23].

Agam Gupta

Member since Sep 21, 2013

Given ratio of sum of n terms of two APâ€™s = (7n+1):(4n+27) Letâ€™s consider the ratio these two APâ€™s mth terms as am : aâ€™m â†’(2) Recall the nth term of AP formula, an = a + (n â€“ 1)d Hence equation (2) becomes, am : aâ€™m = a + (m â€“ 1)d : aâ€™ + (m â€“ 1)dâ€™ On multiplying by 2, we get am : aâ€™m = [2a + 2(m â€“ 1)d] : [2aâ€™ + 2(m â€“ 1)dâ€™] = [2a + {(2m â€“ 1) â€“ 1}d] : [2aâ€™ + {(2m â€“ 1) â€“ 1}dâ€™] = S2m â€“ 1 : Sâ€™2m â€“ 1  = [7(2m â€“ 1) + 1] : [4(2m â€“ 1) +27] [from (1)] = [14m â€“ 7 +1] : [8m â€“ 4 + 27] = [14m â€“ 6] : [8m + 23] Thus the ratio of mth terms of two APâ€™s is [14m â€“ 6] : [8m + 23].

Syeda

Member since Jan 25, 2017

Answer. Given ratio of sum of n terms of two APâ€™s = (7n+1):(4n+27) Letâ€™s consider the ratio these two APâ€™s mth terms as am : aâ€™m â†’(2) Recall the nth term of AP formula, an = a + (n â€“ 1)d Hence equation (2) becomes, am : aâ€™m = a + (m â€“ 1)d : aâ€™ + (m â€“ 1)dâ€™ On multiplying by 2, we get am : aâ€™m = [2a + 2(m â€“ 1)d] : [2aâ€™ + 2(m â€“ 1)dâ€™] = [2a + {(2m â€“ 1) â€“ 1}d] : [2aâ€™ + {(2m â€“ 1) â€“ 1}dâ€™] = S2m â€“ 1 : Sâ€™2m â€“ 1 = [7(2m â€“ 1) + 1] : [4(2m â€“ 1) +27] [from (1)] = [14m â€“ 7 +1] : [8m â€“ 4 + 27] = [14m â€“ 6] : [8m + 23] Thus the ratio of mth terms of two APâ€™s is [14m â€“ 6] : [8m + 23].

Dinesh Singha

Given ratio of sum of n terms of two APâ€™s = (7n+1):(4n+27) Letâ€™s consider the ratio these two APâ€™s mth terms as am : aâ€™m â†’(2) Recall the nth term of AP formula, an = a + (n â€“ 1)d Hence equation (2) becomes, am : aâ€™m = a + (m â€“ 1)d : aâ€™ + (m â€“ 1)dâ€™ On multiplying by 2, we get am : aâ€™m = [2a + 2(m â€“ 1)d] : [2aâ€™ + 2(m â€“ 1)dâ€™] = [2a + {(2m â€“ 1) â€“ 1}d] : [2aâ€™ + {(2m â€“ 1) â€“ 1}dâ€™] = S2m â€“ 1 : Sâ€™2m â€“ 1  = [7(2m â€“ 1) + 1] : [4(2m â€“ 1) +27] [from (1)] = [14m â€“ 7 +1] : [8m â€“ 4 + 27] = [14m â€“ 6] : [8m + 23] Thus the ratio of mth terms of two APâ€™s is [14m â€“ 6] : [8m + 23].

Kishore Kumar

Given ratio of sum of n terms of two APâ€™s = (7n+1):(4n+27) Letâ€™s consider the ratio these two APâ€™s mth terms as am : aâ€™m â†’(2) Recall the nth term of AP formula, an = a + (n â€“ 1)d Hence equation (2) becomes, am : aâ€™m = a + (m â€“ 1)d : aâ€™ + (m â€“ 1)dâ€™ On multiplying by 2, we get am : aâ€™m = [2a + 2(m â€“ 1)d] : [2aâ€™ + 2(m â€“ 1)dâ€™] = [2a + {(2m â€“ 1) â€“ 1}d] : [2aâ€™ + {(2m â€“ 1) â€“ 1}dâ€™] = S2m â€“ 1 : Sâ€™2m â€“ 1 = [7(2m â€“ 1) + 1] : [4(2m â€“ 1) +27] [from (1)] = [14m â€“ 7 +1] : [8m â€“ 4 + 27] = [14m â€“ 6] : [8m + 23] Thus the ratio of mth terms of two APâ€™s is [14m â€“ 6] : [8m + 23].
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