The ratio between the sum of n term of two A.P.'s is 3 n+8: 7 n+15. Find the ratio between their

step 1 Hello. So the question is taken from progression and the question is the ratio between the sum o

The ratio between the sum of n term of two A.P.'s is 3 n+8:...

Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This constant, known as the common difference, makes it possible to express any term in the sequence in a simple form, and allows for straightforward computation of properties like the nth term and the sum of the first n terms.

Sum of an 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 provides a relationship between the sum, the number of terms, and the sequence’s parameters, which is especially useful when comparing sums of two arithmetic sequences.

Nth Term of an Arithmetic Progression

The nth term of an arithmetic progression is given by T? = a + (n ? 1)d. This expression allows us to determine any term in the sequence once the first term and the common difference are known. It is particularly helpful when finding ratios of corresponding terms from two different arithmetic progressions.

Equating Polynomial Coefficients

When an identity holds for all values of a variable, the coefficients of the corresponding powers in the polynomials on both sides must be equal. This method is used to set up equations that equate the parameters (like the first term and common difference) when given a condition that must hold for an arithmetic progression, such as when comparing the sum formulas of two sequences.

Ratio and Proportions

Ratios and proportions are used to compare quantities and are instrumental in problems involving arithmetic progressions when given relationships between sums or terms. By setting up proportional relationships between the sums of sequences or their respective terms, we can solve for unknown parameters like the common difference or the first term.

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