Why is it useful to have an explicit formula for the nth term of a sequence?
Step-by-step solution
Idea: An explicit formula links a term to its position. You can go from position to term (substitute n) and from term to position (solve for n).
- Any term directly: for the odd numbers un = 2n − 1, the 300th term is 2 × 300 − 1 = 599. We did not need the 299 terms before it.1 mark
- Checking a number: to see whether 137 is an odd number in the sequence, solve 2n − 1 = 137: 2n = 138, n = 69. So 137 is the 69th term. If n is not a natural number, the number is not a term.1 mark
Answer to write in the exam
1. Any term is found directly from its position, without the earlier terms.
e.g. 300th odd number = 2 × 300 − 1 = 599
2. It tells whether a number is a term and its position.
e.g. 2n − 1 = 137 ⇒ n = 69 ⇒ 137 is the 69th term
Common mistakes that cost marks
- Saying only ‘it is quicker’. Say exactly what is quicker: finding a far-away term without listing the earlier terms.
- Forgetting the second use: solving the formula for n tells you whether a number belongs to the sequence and where.
- Thinking every sequence has a simple explicit formula. Some, like the prime numbers, do not.
How this can come in the exam
Assertion (A): Using un = 2n − 1, the 500th odd number can be found without listing the first 499.
Reason (R): An explicit formula gives a term directly from its position n.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
u500 = 999 straight from the formula. R explains A.
Try one yourself
For sn = 4n + 1, find the 250th term, and decide whether 81 is a term.
Show answer
s250 = 1001. 4n + 1 = 81 gives n = 20, so 81 is the 20th term.
More questions like this
- Using the explicit rule un = 2n − 1, find the 53rd term, the 108th term, and the 1170th term of the odd number sequence.
- Consider the sequence that is generated by the explicit formula sn = 5n − 2.
- Can you find the rule describing the nth term of the sequence of square numbers?
- Here is the sequence of the first ten prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Do you see any pattern in this sequence? Can you think of a rule that can predict the next few prime numbers?
- Consider the expression tn = 3n − 7.
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