The ratio of the sum of n terms of two arithmetic...

The ratio of the sum of $n$ terms of two arithmetic progressions is given by $(2 n+3):(5 n-7)$. Find the ratio of their $\mathrm{nth}$ terms.
(1) $(4 n+5):(10 n+2)$
(2) $(4 n+1):(10 n-12)$
(3) $(4 n-1):(10 n+8)$
(4) $(4 n-5):(10 n-2)$

Key Concepts

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This property makes the arithmetic progression predictable and allows for the derivation of formulas for both 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 typically 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 problems involving cumulative totals or comparisons of sums over a certain number of terms.

Nth Term of an Arithmetic Progression

The nth term of an arithmetic progression is given by a? = a + (n - 1)d. This formula helps in identifying the individual term at any position in the sequence, which is useful in problems that require an analysis of term-by-term behavior rather than cumulative sums.

Ratio of Quantities

In mathematical problems, establishing the ratio of different quantities helps to compare their relative magnitudes. When applied to arithmetic progressions, ratios can be used to compare sums, individual terms, or other related expressions, often leading to equations that can be solved by equating coefficients.

Equating Coefficients

Equating coefficients is a method where corresponding coefficients of similar terms from two algebraic expressions are set equal to each other. This technique is crucial for identifying relationships between parameters—such as the first term and common difference—in sequences or in situations involving ratios and sums.

Transcript

Please give Ace some feedback