GRE Math : How to find the nth term of an arithmetic sequence

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Example Question #2 : Sequences

The first term in a sequence of integers is 2 and the second term is 10. All subsequent terms are the arithmetic mean of all of the preceding terms. What is the 39th term?

Possible Answers:

1200

600

300

Correct answer:

Explanation:

The first term and second term average out to 6. So the third term is 6. Now add 6 to the preceding two terms and divide by 3 to get the average of the first three terms, which is the value of the 4th term. This, too, is 6 (18/3)—all terms after the 2nd are 6, including the 39th. Thus, the answer is 6.

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Example Question #1 : Arithmetic Sequences

Consider the following sequence of integers:

5, 11, 23, 47

What is the 6^th element in this sequence?

Possible Answers:

191

None of the other answers

189

Correct answer:

191

Explanation:

First, consider the change in each element. Notice that in each case, a given element is twice the preceding one plus one:

11 = 2 * 5 + 1

23 = 11 * 2 + 1

47 = 23 * 2 + 1

To find the 6^th element, continue following this:

The 5^th: 47 * 2 + 1 = 95

The 6^th: 95 * 2 + 1 = 191

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Example Question #2 : Sequences

The sequence $\small a_{1}+a_{2}+... +a_{n}$ $\displaystyle \small a_{1}+a_{2}+... +a_{n}$ begins with the numbers $\displaystyle 3, 11, 18, . . .$ and has the $\small n^{th}$ $\displaystyle \small n^{th}$ term defined as $\displaystyle a_1+2n+n^2$ , for $\small n\geq 2$ $\displaystyle \small n\geq 2$ .

What is the value of the $20^{th}$ $\displaystyle 20^{th}$ term of the sequence?

Possible Answers:

220 $\displaystyle 220$

460 $\displaystyle 460$

443 $\displaystyle 443$

155 $\displaystyle 155$

163 $\displaystyle 163$

Correct answer:

443 $\displaystyle 443$

Explanation:

The first term of the sequence is $\small a_{1}$ $\displaystyle \small a_{1}$ , so here $\small a_{1} = 3$ $\displaystyle \small a_{1} = 3$ , and we're interested in finding the 20th term, so we'll use n = 20.

Plugging these values into the given expression for the nth term gives us our answer.

$\displaystyle a_1+2n+n^2$

$\small a_{1} = 3$ $\displaystyle \small a_{1} = 3$ and n=20 $\displaystyle n=20$

$\small 3 + 2(20) + 20^{2} = 443$ $\displaystyle \small 3 + 2(20) + 20^{2} = 443$

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Example Question #3 : Sequences

In a sequence of numbers, the first two values are 1 and 2. Each successive integer is calculated by adding the previous two and mutliplying that result by 3. What is fifth value in this sequence?

Possible Answers:

126 $\displaystyle 126$

$\displaystyle 39$

129 $\displaystyle 129$

None of the other answers

$\displaystyle 33$

Correct answer:

126 $\displaystyle 126$

Explanation:

Our sequence begins as 1, 2.

Element 3: (Element 1 + Element 2) * 3 = (1 + 2) * 3 = 3 * 3 = 9

Element 4: (Element 2 + Element 3) * 3 = (2 + 9) * 3 = 11 * 3 = 33

Element 5: (Element 3 + Element 4) * 3 = (9 + 33) * 3 = 42 * 3 = 126

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Example Question #1 : Nth Term Of An Arithmetic Sequence

Let Z represent a sequence of numbers $\displaystyle (z_1,z_2,z_3,z_4,...,z_n)$ wherein each term is defined as seven less than three times the preceding term. If $\displaystyle z_3+z_5=142$ , what is the first term in the sequence?

Possible Answers:

$\displaystyle 5$

$\displaystyle 44$

$\displaystyle 17$

125 $\displaystyle 125$

$\displaystyle 8$

Correct answer:

$\displaystyle 5$

Explanation:

Let us first write the value of a consecutive term in a numerical format:

$z_{n+1} = 3z_n-7$ $\displaystyle z_{n+1} = 3z_n-7$

Consequently,

$z_n=\frac{z_{n+1}+7}{3}$ $\displaystyle z_n=\frac{z_{n+1}+7}{3}$

Using the first equation, we can define z_5 $\displaystyle z_5$ in terms of z_3 $\displaystyle z_3$ :

$\displaystyle z_5=3z_4-7=3(3z_3-7)-7=9z_3-21-7=9z_3-28$

This allows us to rewrite

$\displaystyle z_3+z_5=142$

$\displaystyle z_3+9z_3-28=142$

Rearrangement of terms allows us to solve for z_3 $\displaystyle z_3$ :

$\displaystyle 10z_3=170$

z_3=17 $\displaystyle z_3=17$

Now, using our second equation, we can find z_1 $\displaystyle z_1$ , the first term:

$z_{1}=\frac{z_2+7}{3}=\frac{\frac{z_{3}+7}{3}+7}{3}=\frac{\frac{24}{3}+7}{3}=\frac{15}{3}=5$ $\displaystyle z_{1}=\frac{z_2+7}{3}=\frac{\frac{z_{3}+7}{3}+7}{3}=\frac{\frac{24}{3}+7}{3}=\frac{15}{3}=5$

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GRE Math : How to find the nth term of an arithmetic sequence

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

Example Question #2 : Sequences

Example Question #1 : Arithmetic Sequences

Example Question #2 : Sequences

Example Question #3 : Sequences

Example Question #1 : Nth Term Of An Arithmetic Sequence

All GRE Math Resources

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