To find the ratio of the sum of m terms for two arithmetic progressions (APs) given that the ratio of their sums for n terms is 7n, 14n, and 27, we can use the properties of arithmetic progressions.
The sum of the first n terms of an AP can be expressed as:
Sn = n/2 * (2a + (n-1)d)
Where a is the first term and d is the common difference. For two APs, we’ll denote their sums as:
S1(n) = 7n
S2(n) = 14n
To find the ratio for m terms, we extend the formula for m terms:
S1(m) = m/2 * (2a1 + (m-1)d1)
S2(m) = m/2 * (2a2 + (m-1)d2)
The ratio of the sums for m terms will be:
Ratio = S1(m) : S2(m) = (7/14) * (m/2)(2a1 + (m-1)d1) : (m/2)(2a2 + (m-1)d2)
Since we know the ratio of the sums for n terms, the relationship holds. Thus, the ratio of the sums of m terms remains consistent with what we observed for n terms. Therefore:
The ratio of the sums of the m terms of the two APs is 7 : 14, which simplifies down to a final ratio of:
1 : 2
Thus, for m terms, the ratio of the sums of the two arithmetic progressions is 1 : 2.