Instructions
The 'nth term' is a formula that allows you to find any term in a sequence. For a linear sequence (one that has a common difference), follow these steps:
- Find the Common Difference: Look at how much the sequence goes up or down by each time. This number goes in front of 'n'.
- Write the Times Table: Write down the times table for the common difference you just found.
- Compare and Adjust: Compare the times table to your original sequence. Figure out what you need to add or subtract to get from the times table to your sequence. This number goes at the end of your formula.
Example: Find the nth term of the sequence 5, 8, 11, 14, ...
- Step 1: The sequence goes up by 3 each time. So the formula starts with 3n.
- Step 2: The 3 times table is 3, 6, 9, 12, ...
- Step 3: To get from the 3 times table (3, 6, 9) to our sequence (5, 8, 11), we need to add 2 each time.
- Result: The nth term is 3n + 2.
Part 1: Find the nth Term
Find the nth term formula for each of the following sequences.
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8, 10, 12, 14, 16, ...
nth term:
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5, 9, 13, 17, 21, ...
nth term:
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20, 17, 14, 11, 8, ...
nth term:
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1, 6, 11, 16, 21, ...
nth term:
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-2, 0, 2, 4, 6, ...
nth term:
Part 2: Use the nth Term
Use the given nth term formula to find the requested terms in each sequence.
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The nth term is 6n + 1.
- What is the 5th term?
- What is the 10th term?
- What is the 100th term?
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The nth term is 20 - 3n.
- What is the 1st term?
- What is the 4th term?
- What is the 10th term?
Part 3: Problem Solving
Use the nth term to answer the following question. Show your working.
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The nth term of a sequence is 4n - 3. Is the number 57 a term in this sequence?
(Hint: Set the formula equal to 57 and solve for n. If n is a whole number, then it is in the sequence).
Answer Key
Part 1: Find the nth Term
- 2n + 6 (Goes up in 2s. The 2 times table is 2, 4, 6... need to add 6)
- 4n + 1 (Goes up in 4s. The 4 times table is 4, 8, 12... need to add 1)
- -3n + 23 (Goes down in 3s. The -3 times table starts with -3... need to add 23 to get to 20)
- 5n - 4 (Goes up in 5s. The 5 times table is 5, 10, 15... need to subtract 4)
- 2n - 4 (Goes up in 2s. The 2 times table is 2, 4, 6... need to subtract 4)
Part 2: Use the nth Term
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The nth term is 6n + 1.
- 5th term: (6 x 5) + 1 = 31
- 10th term: (6 x 10) + 1 = 61
- 100th term: (6 x 100) + 1 = 601
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The nth term is 20 - 3n.
- 1st term: 20 - (3 x 1) = 17
- 4th term: 20 - (3 x 4) = 20 - 12 = 8
- 10th term: 20 - (3 x 10) = 20 - 30 = -10
Part 3: Problem Solving
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The nth term of a sequence is 4n - 3. Is the number 57 a term in this sequence?
4n - 3 = 57
4n = 57 + 3
4n = 60
n = 60 / 4
n = 15
Yes, 57 is the 15th term in the sequence.