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Finding the step, filling a gap, and using an nth term rule to jump ahead.
Question 1 of 12
A sequence starts 10, 12, 14, 16, ... The rule for the nth term is 2n + 8. What is the 50th term?
The rule works for any position, so there is no need to continue the sequence term by term. Put n = 50 into 2n + 8: 2 × 50 = 100, then add 8, giving 108. Checking the rule on a term you can see is always worth doing first: n = 1 gives 2 × 1 + 8 = 10, which matches the sequence. The number in front of n is always the step between terms. Spotting it is most of the work.
The answer is D. The rule works for any position, so there is no need to continue the sequence term by term. Put n = 50 into 2n + 8: 2 × 50 = 100, then add 8, giving 108. Checking the rule on a term you can see is always worth doing first: n = 1 gives 2 × 1 + 8 = 10, which matches the sequence. The number in front of n is always the step between terms. Spotting it is most of the work.
A sequence starts 19, 28, 37, 46, ... The rule for the nth term is 9n + 10. What is the 50th term?
The rule works for any position, so there is no need to continue the sequence term by term. Put n = 50 into 9n + 10: 9 × 50 = 450, then add 10, giving 460. Checking the rule on a term you can see is always worth doing first: n = 1 gives 9 × 1 + 10 = 19, which matches the sequence. The nth term rule exists precisely so you do not have to count all the way there.
The answer is B. The rule works for any position, so there is no need to continue the sequence term by term. Put n = 50 into 9n + 10: 9 × 50 = 450, then add 10, giving 460. Checking the rule on a term you can see is always worth doing first: n = 1 gives 9 × 1 + 10 = 19, which matches the sequence. The nth term rule exists precisely so you do not have to count all the way there.
What is the next number in this sequence? 5, 13, 21, 29, ...
Look at the gaps between the terms rather than at the numbers themselves. 13 − 5 = 8, and the same gap appears each time, so the rule is add 8 each step. 29 + 8 = 37. A sequence going down has a negative step. The method does not change.
The answer is C. Look at the gaps between the terms rather than at the numbers themselves. 13 − 5 = 8, and the same gap appears each time, so the rule is add 8 each step. 29 + 8 = 37. A sequence going down has a negative step. The method does not change.
A sequence starts 10, 19, 28, 37, ... The rule for the nth term is 9n + 1. What is the 10th term?
The rule works for any position, so there is no need to continue the sequence term by term. Put n = 10 into 9n + 1: 9 × 10 = 90, then add 1, giving 91. Checking the rule on a term you can see is always worth doing first: n = 1 gives 9 × 1 + 1 = 10, which matches the sequence. Test the rule on the first term before trusting it on the fiftieth.
The answer is C. The rule works for any position, so there is no need to continue the sequence term by term. Put n = 10 into 9n + 1: 9 × 10 = 90, then add 1, giving 91. Checking the rule on a term you can see is always worth doing first: n = 1 gives 9 × 1 + 1 = 10, which matches the sequence. Test the rule on the first term before trusting it on the fiftieth.
What is the next number in this sequence? 1, 8, 15, 22, ...
Look at the gaps between the terms rather than at the numbers themselves. 8 − 1 = 7, and the same gap appears each time, so the rule is add 7 each step. 22 + 7 = 29. Write the gaps above the arrows between the terms. Once they are visible the rule usually is too.
The answer is D. Look at the gaps between the terms rather than at the numbers themselves. 8 − 1 = 7, and the same gap appears each time, so the rule is add 7 each step. 22 + 7 = 29. Write the gaps above the arrows between the terms. Once they are visible the rule usually is too.
A sequence starts 9, 18, 27, 36, ... The rule for the nth term is 9n. What is the 15th term?
The rule works for any position, so there is no need to continue the sequence term by term. Put n = 15 into 9n: 9 × 15 = 135, so the 15th term is 135. Checking the rule on a term you can see is always worth doing first: n = 1 gives 9 × 1 + 0 = 9, which matches the sequence. The number in front of n is always the step between terms. Spotting it is most of the work.
The answer is C. The rule works for any position, so there is no need to continue the sequence term by term. Put n = 15 into 9n: 9 × 15 = 135, so the 15th term is 135. Checking the rule on a term you can see is always worth doing first: n = 1 gives 9 × 1 + 0 = 9, which matches the sequence. The number in front of n is always the step between terms. Spotting it is most of the work.
A sequence starts 14, 17, 20, 23, ... The rule for the nth term is 3n + 11. What is the 100th term?
The rule works for any position, so there is no need to continue the sequence term by term. Put n = 100 into 3n + 11: 3 × 100 = 300, then add 11, giving 311. Checking the rule on a term you can see is always worth doing first: n = 1 gives 3 × 1 + 11 = 14, which matches the sequence. The nth term rule exists precisely so you do not have to count all the way there.
The answer is C. The rule works for any position, so there is no need to continue the sequence term by term. Put n = 100 into 3n + 11: 3 × 100 = 300, then add 11, giving 311. Checking the rule on a term you can see is always worth doing first: n = 1 gives 3 × 1 + 11 = 14, which matches the sequence. The nth term rule exists precisely so you do not have to count all the way there.
What number is missing from this sequence? 5, 9, 13, ?, 21
The gap cannot be read straight off either side of the hole, but it can be read elsewhere: 21 − 17 = 4 and 13 − 9 = 4, so the sequence goes up in 4s. The missing term is 13 + 4 = 17. It can be checked from the other side too: 21 − 4 = 17. The gap either side of a missing term is double the step. That catches people out.
The answer is C. The gap cannot be read straight off either side of the hole, but it can be read elsewhere: 21 − 17 = 4 and 13 − 9 = 4, so the sequence goes up in 4s. The missing term is 13 + 4 = 17. It can be checked from the other side too: 21 − 4 = 17. The gap either side of a missing term is double the step. That catches people out.
What is the next number in this sequence? 15, 10, 5, 0, ...
Look at the gaps between the terms rather than at the numbers themselves. 10 − 15 = -5, and the same gap appears each time, so the rule is subtract 5 each step. 0 − 5 = -5. A sequence going down has a negative step. The method does not change.
The answer is B. Look at the gaps between the terms rather than at the numbers themselves. 10 − 15 = -5, and the same gap appears each time, so the rule is subtract 5 each step. 0 − 5 = -5. A sequence going down has a negative step. The method does not change.
What is the next number in this sequence? 10, 13, 16, 19, ...
Look at the gaps between the terms rather than at the numbers themselves. 13 − 10 = 3, and the same gap appears each time, so the rule is add 3 each step. 19 + 3 = 22. Write the gaps above the arrows between the terms. Once they are visible the rule usually is too.
The answer is A. Look at the gaps between the terms rather than at the numbers themselves. 13 − 10 = 3, and the same gap appears each time, so the rule is add 3 each step. 19 + 3 = 22. Write the gaps above the arrows between the terms. Once they are visible the rule usually is too.
A sequence starts 5, 14, 23, 32, ... The rule for the nth term is 9n − 4. What is the 50th term?
The rule works for any position, so there is no need to continue the sequence term by term. Put n = 50 into 9n − 4: 9 × 50 = 450, then subtract 4, giving 446. Checking the rule on a term you can see is always worth doing first: n = 1 gives 9 × 1 − 4 = 5, which matches the sequence. Test the rule on the first term before trusting it on the fiftieth.
The answer is A. The rule works for any position, so there is no need to continue the sequence term by term. Put n = 50 into 9n − 4: 9 × 50 = 450, then subtract 4, giving 446. Checking the rule on a term you can see is always worth doing first: n = 1 gives 9 × 1 − 4 = 5, which matches the sequence. Test the rule on the first term before trusting it on the fiftieth.
What number is missing from this sequence? 9, 16, 23, ?, 37
The gap cannot be read straight off either side of the hole, but it can be read elsewhere: 37 − 30 = 7 and 23 − 16 = 7, so the sequence goes up in 7s. The missing term is 23 + 7 = 30. It can be checked from the other side too: 37 − 7 = 30. The gap either side of a missing term is double the step. That catches people out.
The answer is D. The gap cannot be read straight off either side of the hole, but it can be read elsewhere: 37 − 30 = 7 and 23 − 16 = 7, so the sequence goes up in 7s. The missing term is 23 + 7 = 30. It can be checked from the other side too: 37 − 7 = 30. The gap either side of a missing term is double the step. That catches people out.
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