The sum of the first n terms of two arithmetic progressions...

The sum of the first $n$ terms of two arithmetic progressions $S_{1}$ and $S_{2}$ are in the ratio $11 n-17: 5 n-21 .$ Find the ratio of the $16^{\text {th }}$ terms of $S$, and $S_{2}$.
(A) $3: 2$
(B) $162: 67$
(C) $9: 4$
(D) $27: 8$

Key Concepts

Relation Between Term and Sum

The nth term of an arithmetic progression can be derived from its sum formula and is given by a_n = a + (n - 1)d. Understanding this link is important because it connects the cumulative properties of the sequence with the individual terms, thereby enabling the analysis of ratios between specific terms of different sequences.

Ratio Comparisons in Sequences

Comparing ratios either of sums or individual terms between two arithmetic progressions involves aligning their formulas and coefficients appropriately. This process often leads to equating expressions to find relationships between their first terms and common differences, which can then be used to derive specific term ratios.

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which the difference between any two successive terms is constant, known as the common difference. This fundamental concept provides a basis for analyzing linear relationships between terms and is widely used in various areas of mathematics, including algebra and number theory.

Sum Formula for an Arithmetic Progression

The sum of the first n terms of an arithmetic progression is given by the formula S_n = n/2 × [2a + (n - 1)d], where 'a' is the first term and 'd' is the common difference. This formula allows for the calculation of the aggregate value of a set number of terms and is key when establishing relationships between different arithmetic progressions.

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