If the sum of first n terms of two A.P's are in the ratio 3 n+8: 7 n+15, then the ratio of their

step 1 we know that the sum of the first N terms of two arithmetic progression are in the ratio of 3N p

If the sum of first n terms of two A.P's are in the ratio 3...

Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This common difference allows the sequence to be expressed in a structured way using simple formulas, and it forms the basis for deriving other properties such as individual terms and the sum of terms.

Sum Formula for Arithmetic Progression

The sum of the first n terms of an arithmetic progression is given by the formula S? = n/2 [2a + (n - 1)d], where a is the first term and d is the common difference. This formula is essential for analyzing how the sum behaves as the number of terms increases and is useful in problems that compare the sums of different sequences.

Term Extraction from Sum Function

The nth term of an arithmetic progression can be extracted from its sum function using the relationship a? = S? - S???. This method is particularly valuable when the sum expression is known, as it helps to isolate individual term values from cumulative sums.

Coefficient Comparison in Ratios

When working with ratios of expressions that involve a variable (such as n), comparing the coefficients of like powers of n in the expressions simplifies the relationship between the parameters of the sequences. This technique facilitates the determination of individual term ratios or other key relationships by removing the dependency on the variable.

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