9.2 Arithmetic & Geometric Sequences

Arithmetic & Geometric Sequences

Arithmetic and Geometric Sequences is one of the eight named topics in the NTS Algebra block. Sequence items reward pure formula recall more than almost anything else on the quantitative paper: four formulas cover the entire topic, and once you have them, most items resolve in under a minute.

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Telling the Two Apart

Given a list of numbers, test both operations on consecutive terms:

• $4,;11,;18,;25,;\ldots$ — differences are all 7 → arithmetic.
• $3,;12,;48,;192,;\ldots$ — ratios are all 4 → geometric.
• $2,;6,;24,;120,;\ldots$ — neither; these are factorial-style products $\times 3, \times 4, \times 5$. Some NTS items are deliberately neither, and the answer is found by pattern description rather than by formula.

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Arithmetic Sequences

$a_n = a + (n-1)d \qquad\qquad S_n = \frac{n}{2}\bigl[2a + (n-1)d\bigr] = \frac{n}{2}(a + a_n)$

The second form of the sum is the one to reach for whenever you already know the last term.

Worked Example — Finding a Term

The 5th term of an AP is 23 and the 12th term is 58. Find the 20th term.

$a + 4d = 23 \qquad a + 11d = 58$

Subtracting: $7d = 35$, so $d = 5$ and $a = 23 - 20 = 3$.

$a_{20} = 3 + 19(5) = 98$

Worked Example — Summing

Sum all multiples of 6 between 20 and 200.

First multiple is 24, last is 198, difference 6.

$n = \frac{198 - 24}{6} + 1 = \frac{174}{6} + 1 = 30 \qquad S_{30} = \frac{30}{2}(24 + 198) = 15 \times 222 = 3{,}330$

The plus one in the count formula is the most-forgotten step in the whole topic.

Arithmetic Mean

The AM of $x$ and $y$ is $\dfrac{x+y}{2}$. To insert $k$ arithmetic means between $x$ and $y$, treat the result as an AP with $k+2$ terms, so $d = \dfrac{y-x}{k+1}$.

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Geometric Sequences

$a_n = a\,r^{\,n-1} \qquad\qquad S_n = \frac{a(r^{n} - 1)}{r - 1}\;\;(r \neq 1) \qquad\qquad S_\infty = \frac{a}{1 - r}\;\;\bigl(|r| < 1\bigr)$

Worked Example — Finding a Term

The 3rd term of a GP is 20 and the 6th term is 160. Find the first term.

$\frac{ar^{5}}{ar^{2}} = \frac{160}{20} = 8 \quad\Rightarrow\quad r^{3} = 8 \quad\Rightarrow\quad r = 2$ $ar^{2} = 20 \quad\Rightarrow\quad 4a = 20 \quad\Rightarrow\quad a = 5$

The sequence is $5,;10,;20,;40,;80,;160,\ldots$ ✓

Worked Example — Infinite Series

Sum $18 + 6 + 2 + \dfrac{2}{3} + \cdots$

Here $a = 18$ and $r = \frac{1}{3}$. Since $|r| < 1$ the series converges:

$S_\infty = \frac{18}{1 - \frac{1}{3}} = \frac{18}{\frac{2}{3}} = 27$

If $|r| \ge 1$ the series diverges and has no finite sum. An option offering a finite value for a series with $r = 2$ is always wrong.

Geometric Mean

The GM of two positive numbers is $\sqrt{xy}$. The GM of 4 and 25 is $\sqrt{100} = 10$, whereas their AM is 14.5 — for any two distinct positive numbers, $AM > GM$ always.

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Word Problems That Hide a Sequence

Worked Word Problem

A machine bought for Rs. 800,000 depreciates 25% each year. What is its value after 3 years?

Each year retains 75%, so this is a GP with $r = 0.75$:

$V_3 = 800000 \times (0.75)^{3} = 800000 \times 0.421875 = \text{Rs. }337{,}500$

The trap answer is $800000 - (3 \times 200000) = 200{,}000$, which wrongly treats percentage depreciation as a fixed rupee deduction — arithmetic thinking applied to a geometric process.

Twenty-second routine: compute the second-minus-first difference and the second-divided-by-first ratio. Check both against the next pair. Whichever stays constant tells you which pair of formulas to use, and you have solved the classification step before reading the question.