Write the first five terms of the sequence whose nth term is aₙ = 2ⁿ.

Exercise 8.1 · Q3

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Concept understanding — Sequence Terms Evaluation

Sequence Terms Evaluation

A parking garage numbers its levels 1, 2, 3, ... and posts a rule on the wall: "Level $n$ sits $n^2$ metres below street level." You don't need to walk down to level 10 to find its depth -- you simply compute $10^2 = 100$ metres. That one substitution is the entire idea behind sequence terms evaluation: once a sequence has a rule connecting position to value, any term can be found directly, without listing every term before it.

What "Evaluating a Term" Means

A sequence assigns a number to each position $1, 2, 3, \dots$ . The term at position $n$ is written $a_n$ (read "a sub n"). When the sequence is given by an explicit formula $a_n = f(n)$ , evaluating the $k$ -th term simply means substituting $n = k$ into $f$ and simplifying.

Note

This is different from a recursive definition, where $a_n$ is defined using $a_{n-1}$ -- for example, $a_1 = 1,\ a_n = a_{n-1} + 2$ . A recursive rule forces you to compute every earlier term first; an explicit formula lets you jump straight to any term you want.

The Precise Statement

If $\{a_n\}$ is a sequence with $a_n = f(n)$ , then for any positive integer $k$ , the $k$ -th term is

a_k = f(k)

The entire procedure is: replace $n$ with the required index, then simplify the resulting number. No equation-solving, no guesswork -- just substitution.

Common Forms You'll Meet in Exams

Type	Formula	5th term
Linear	$a_n = 3n - 2$	$3(5)-2 = 13$
Quadratic	$a_n = n^2 + 1$	$5^2+1 = 26$
Exponential	$a_n = 2^n$	$2^5 = 32$
Rational	$a_n = \dfrac{n}{n+1}$	$\dfrac{5}{6}$
Alternating	$a_n = (-1)^n$	$(-1)^5 = -1$

Worked Example

Find the 7th term of the sequence $a_n = \dfrac{3n-1}{n^2+2}$ .

The index we need is $k = 7$ .
Substitute $n = 7$ :

a_7 = \frac{3(7)-1}{7^2+2} = \frac{21-1}{49+2} = \frac{20}{51}

Check whether it simplifies -- $\gcd(20, 51) = 1$ , so $\dfrac{20}{51}$ is already in lowest terms and is the final answer.

Watch out

A common mistake is starting from the wrong index. Unless a problem says otherwise, sequences here start at $n = 1$ , not $n = 0$ . For $a_n = n - 3$ , the correct first term is $a_1 = 1 - 3 = -2$ . A student who mistakenly plugs in $n = 0$ instead gets $-3$ -- a different, wrong number, obtained by starting one position too early. Always check where the sequence is defined to begin.

Tip

When the formula has fractions or exponents, write the substitution step out fully before simplifying -- it prevents sign and arithmetic slips and shows the examiner exactly how you reached the answer.

Why This Matters

Sequence term evaluation is the foundation on which everything else in this chapter is built:

Finding a specific term directly, without listing the whole sequence.
Checking membership -- set $a_n$ equal to a target number and solve for $n$ to see if that number appears in the sequence.
Feeding sums -- evaluating individual terms is the first step before adding several of them together, as in arithmetic and geometric series.
Comparing behaviour -- evaluating terms for large $n$ hints at whether a sequence grows, shrinks, or settles down.

Master substitution into an explicit formula first -- arithmetic progressions, geometric progressions, and every special series in this chapter reuse this exact skill, just with more specific formulas for $a_n$ .

Substituting $n=1,2,3,4,5$ into $a_n=2^n$ gives the first five terms: $2, 4, 8, 16, 32$ .

A sequence given by an explicit formula lets us find any term by direct substitution. Here $a_n = 2^n$ means each term is a power of 2, with the exponent equal to the term's position.

First term ( $n=1$ ): $a_1 = 2^1 = 2$
Second term ( $n=2$ ): $a_2 = 2^2 = 4$
Third term ( $n=3$ ): $a_3 = 2^3 = 8$
Fourth term ( $n=4$ ): $a_4 = 2^4 = 16$
Fifth term ( $n=5$ ): $a_5 = 2^5 = 32$

Notice each term is double the previous one — this is a geometric sequence with common ratio $r=2$ .

Tip

For $a_n = r^n$ , each term is obtained by multiplying the previous term by $r$ ; recognizing this speeds up calculation and helps catch errors.

$n$	$a_n = 2^n$
1	2
2	4
3	8
4	16
5	32

✓Final answer

2, 4, 8, 16, 32

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Q.Write the first five terms of the sequence whose nnnth term is an=2na_n = 2^nan​=2n.

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Q.Write the first five terms of the sequence whose $n$ th term is $a_n = 2^n$ .

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