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A. Many times instead of finding the next term in a sequence you are asked to find a specific term in a sequence. For problems like this use the following:
1. When asked this question, almost all the sequences will be adding a fixed number to each successive term. (See Finding The Next Term part C)
2. This problem I always approach from a logic standpoint.
3. Ask yourself, what should the first term be? (This is 'a'.) This is always equal to the fixed number that is being added.
4. Now, how can I change the number in step 3 (by adding or subtracting a number) to equal the first term? (This is 'b'.)
5. Now, make an expression for the nth term using the above information:
an + b
Ex  Find the 11th term of the sequence 4, 7, 10,...
a. Notice that 3 is being added to all numbers, so the first term should be 3, then 6, then 9, etc.
b. You need to add 1 to 3 to equal 4.
c. So the expression is 3n + 1.
d. The answer is 3(11) + 1 or 34.
Ex  Find the 8th term of the sequence -2, 0, 2,...
a. In this case 'a' is 2. So normally we would count: 2,4,6,etc. So the first term should be 2.
b. If we subtract 4 from 2 we get -2.
c. So the expression is 2n - 4.
d. The answer is 2(8) - 4 or 12.
Ex  Find the 40th term in the sequence 1, 7, 13,...
a. In this case 'a' is 6. So the first term should be 6.
b. If we subtract 5 from 6 we get 1.
c. So the expression is 6n - 5.
d. The answer is 6(40) - 5 or 235.
B. Sometimes the question is turned around and asks what term a certain number is in a sequence.
1. We can use the same logic as above to come up with an expression.
2. Find the expression for the nth term of the sequence (same as above).
3. Solve for n: an + b = x, where x is the given number.
Ex  58 is what term in the sequence 1, 4, 7,...
a. Since we are adding by 3, 3 should be the first number.
b. If we subtract 2 from 3 we get 1.
c. So the expression is 3n - 2.
d. Solve: 3n - 2 = 58.
e. The answer is 20.
Ex  105 is what term in the sequence 9, 13, 17,...
a. Since we are adding by 4, the first number should be 4.
b. If we add 5 to 4 we get 9.
c. So the expression is 4n + 5.
d. Solve: 4n + 5 = 105.
e. The answer is 25.
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