Sequence Terms Evaluation
A parking garage numbers its levels 1, 2, 3, ... and posts a rule on the wall: "Level n sits n2 metres below street level." You don't need to walk down to level 10 to find its depth -- you simply compute 102=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,…. The term at position n is written an (read "a sub n"). When the sequence is given by an explicit formula an=f(n), evaluating the k-th term simply means substituting n=k into f and simplifying.
This is different from a recursive definition, where an is defined using an−1 -- for example, a1=1, an=an−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 {an} is a sequence with an=f(n), then for any positive integer k, the k-th term is
ak=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 | an=3n−2 | 3(5)−2=13 |
| Quadratic | an=n2+1 | 52+1=26 |
| Exponential | an=2n | 25=32 |
| Rational | an=n+1n | 65 |
| Alternating | an=(−1)n | (−1)5=−1 |
Worked Example
Find the 7th term of the sequence an=n2+23n−1.
- The index we need is k=7.
- Substitute n=7:
a7=72+23(7)−1=49+221−1=5120
- Check whether it simplifies -- gcd(20,51)=1, so 5120 is already in lowest terms and is the final answer.
A common mistake is starting from the wrong index. Unless a problem says otherwise, sequences here start at n=1, not n=0. For an=n−3, the correct first term is a1=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.
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 an 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 an.