# GRE Subject Test: Math : Finding Terms of a Sequence

## Example Questions

### Example Question #171 : Calculus

If a sequence is 9, 3, -3, -9, -15, ...

What is the 10th term of the sequence?

Explanation:

It will be helpful if you can see that the sequence is changing by -6 to get to each number. The quickest way to solve this is to keep going with the sequence by subtracting six from each number to get to the next number.

9, 3, -3, -9, -15, -21, -27, -33, -39, -45

-45 is the 10th number in the sequence.

Another way to find the 10th term is by using the equation for arithmetic sequences.

where  is the term in question,  is the first term,  is the numbered term, and  is the difference of the terms.

To find d subtract the first two terms.

Therefore, the equation becomes

### Example Question #2 : Finding Terms Of A Sequence

The first 3 terms of a geometric sequence are 5,9, and 13. What is the 15th term?

Explanation:

Step 1: We know it's a geometric sequence, so the difference between each consecutive term is the same. We must find the difference.

Step 2: For every term after the first term, we will add a multiple of  to the first term.

Step 3: We can write an equation that helps us find the nth term of the sequence. We add a multiple of 4n to 5; the equation becomes .

Step 4: We need to find the value of n. We want the 15th number and we are given the first number. There are  numbers in between, so n=14.

Step 5: Plug in 14 for n and find the 15th term

The  term is

### Example Question #1 : Finding Terms Of A Sequence

The first three terms of a arithmetic sequence are . What is the  term of this sequence?

Explanation:

Step 1: We need to find the difference between the terms. To find the difference, subtract the first two terms.

Difference=

Step 2: To find the next term of a arithmetic sequence, we add the difference to the previous term.

Fourth term= term+5
term=. The fourth term is 19.

Step 3: To find the  term, we must add a multiple of  to the first term. In this problem, we are given the  term, so we need to find how many terms are in between the  and  term.

. This tells me that I have to add  fourteen times to get to the  term. .

Step 4: Now that we know what the value of n, we can plug it in to the equation:

, where 4 is the starting number, and n represents how many times I need to add 5 to the next term (and up to the 15th term).

Let's plug it in:

.

So, the  term in this sequence is .

### Example Question #1 : Finding Terms Of A Sequence

What is the next number in the sequence:

Explanation:

Step 1: Try to find a pattern in the sequence

The pattern is decreasing perfect squares, the difference between each base of the consecutive term is ..

Step 2: Find the next term...

If the base of the next term goes down by , the next term is ..

The next term of this sequence is .

### Example Question #1 : Finding Terms Of A Sequence

What is the next term in the sequence:

Explanation:

Step 1: Try to find the difference between the terms.

Difference between  and  is .
Difference between  and  is .
Difference between  and  is .

Difference between  and  is .

The difference between consecutive numbers is an odd number, and the difference between the consecutive terms is always ..

The difference between  and  is

The next number in the sequence is .

### Example Question #1 : Finding Terms Of A Sequence

Find the next term in the sequence:

Explanation:

Step 1: Find the next term in the sequence:

Step 2: Can you recognize the sequence here??

This is the Fibonacci Sequence.. The sum of the previous two terms is equal to the next consecutive term..

The missing term is 13.

### Example Question #1 : Finding Terms Of A Sequence

What is the next term in the sequence:

Explanation:

Step 1: Find the difference between each two consecutive terms...

The difference is always 6 because we have a sequence...

Step 2: Find the next term in the sequence..

The next term is .

### Example Question #2 : Finding Terms Of A Sequence

Find the next term in the following sequence:

Explanation:

Step 1: Calculate the difference between the terms...

Step 2: We found the pattern between the terms..

So, we can add 9 to the term before the  to find the value of ..

The missing term (the next term) in the sequence is .

### Example Question #172 : Calculus

If the first term of an arithmetic sequence is 2 and the third term is 8, find the th term.

Explanation:

Step 1: Find the difference between each term...

Subtract the first term from the third term...

There are two terms between first and third...Take the answer in step 1 and divide by 2 to get the difference between consecutive terms...

The common difference is .

Step 2: Find an equation that describes the sequence....

The equation is , where  represents how many terms I need to calculate and  is the first term...

Step 3: Plug in ...

To find n, we subtract the term that we want from the original term...

So, if we want the th term and we are given the first term...

Then

So,

The th term is .

### Example Question #1 : Finding Terms Of A Sequence

If the first  terms of a sequence are ,find the th term.

Explanation:

Step 1: Find the successive difference rows until we get equal values between every two consecutive numbers.

Step 2: Since we obtain equal values for the successive differences in the second row, hence the  term of the sequence is a second-degree polynomial.

So, the  term takes the form of

Step 3: Now, substituting  into the formula, we get:

Step 4: Solve the system of linear equations:

So, the   term takes the form:

Step 5: Plug in  into the form...