If the ratio of the sum of the first m terms and the first n terms of an AP is m^2: n^2, the rat

step 1 In this problem, I can write the value of SM by SN is equal to m square by n square. On further

If the ratio of the sum of the first m terms and the first n...

If the ratio of the sum of the first $\mathrm{m}$ terms and the first $\mathrm{n}$ terms of an $\mathrm{AP}$ is $\mathrm{m}^{2}: \mathrm{n}^{2}$, the ratio of its $\mathrm{mth}$ and $\mathrm{nth}$ term will be (a) $2 \mathrm{~m}-1: 2 \mathrm{n}-1$ (b) $\mathrm{m}: \mathrm{n}$ (c) $2 \mathrm{~m}+1: 2 \mathrm{n}+1$ (d) None of these

If the ratio of the sum of the first $\mathrm{m}$ terms and the first $\mathrm{n}$ terms of an $\mathrm{AP}$ is $\mathrm{m}^{2}: \mathrm{n}^{2}$, the ratio of its $\mathrm{mth}$ and $\mathrm{nth}$ term will be
(a) $2 \mathrm{~m}-1: 2 \mathrm{n}-1$
(b) $\mathrm{m}: \mathrm{n}$
(c) $2 \mathrm{~m}+1: 2 \mathrm{n}+1$
(d) None of these

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Key Concepts

Arithmetic Progression (AP)

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This common difference is a defining characteristic and allows one to form general formulas for the nth term and for the sum of n terms in the sequence.

Formula for the nth Term of an AP

The nth term of an arithmetic progression is given by the formula a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference. This formula is essential for understanding the position and value of any term within the sequence.

Sum of an Arithmetic Progression

The sum of the first n terms of an arithmetic progression can be calculated using the formula S_n = n/2 * [2a_1 + (n-1)d] or equivalently S_n = n/2 * (a_1 + a_n). This formula is central when solving problems that involve the cumulative total of terms in the sequence over a specified number of terms.

Ratio Reasoning in Sequences

Ratio reasoning in the context of sequences involves comparing quantities derived from an arithmetic progression, such as the ratio of sums or the ratio of specific terms. It requires setting up proportional relationships based on the formulas for the nth term and the sum of terms to deduce information about the sequence or its parameters.

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