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Finding the nth Term Given the Common Difference and a Term

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Arithmetic Sequences and Finding the nth Term Given the Common Difference and a Term
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Halley's Comet appears in the sky approximately every 76 years. The comet was first spotted in the year 1531. Find the n^{th} term rule and the 10^{th} term for the sequence represented by this situation.

Guidance

In this concept we will begin looking at a specific type of sequence called an arithmetic sequence . In an arithmetic sequence the difference between any two consecutive terms is constant. This constant difference is called the common difference . For example, question one in the Review Queue above is an arithmetic sequence. The difference between the first and second terms is (5 - 3) = 2 , the difference between the second and third terms is (7 - 5) = 2 and so on. We can generalize this in the equation below:

a_n-a_{n-1}=d , where a_{n-1} and a_n represent two consecutive terms and d represents the common difference.

Since the same value, the common difference, d , is added to get each successive term in an arithmetic sequence we can determine the value of any term from the first term and how many time we need to add d to get to the desired term as illustrated below:

Given the sequence: 22, 19, 16, 13, \ldots in which a_1=22 and d=-3

a_1&=22 \ or \ 22+(1-1)(-3)=22+0=22 \\a_2&=19 \ or \ 22+(2-1)(-3)=22+(-3)=19 \\a_3&=16 \ or \ 22+(3-1)(-3)=22+(-6)=16 \\a_4&=13 \ or \ 22+(4-1)(-3)=22+(-9)=13 \\&\qquad \qquad \vdots \\a_n&=22+(n-1)(-3) \\a_n&=22-3n+3 \\a_n&=-3n+25

Now we can generalize this into a rule for the n^{th} term of any arithmetic sequence:

a_{n}=a_1+(n-1)d

Example A

Find the common difference and n^{th} term rule for the arithmetic sequence: 2, 5, 8, 11 \ldots

Solution: To find the common difference we subtract consecutive terms.

5-2&=3 \\8-5&=3 \ ,\text{thus the common difference is} \ 3. \\11-8&=3

Now we can put our first term and common difference into the n^{th} term rule discovered above and simplify the expression.

a_n&=2+(n-1)(3) \\&=2+3n-3 \quad ,\text{so} \ a_n=3n-1. \\&=3n-1

Example B

Find the n^{th} term rule and thus the 100^{th} term for the arithmetic sequence in which a_1=-9 and d=2 .

Solution: We have what we need to plug into the rule:

a_n&=-9+(n-1)(2) \\&=-9+2n-2 \quad , \text{thus the} \ n^{th} \ \text{term rule is} \ a_n=2n-11. \\&=2n-11

Now to find the 100^{th} term we can use our rule and replace n with 100: a_{100}=2(100)-11=200-11=189 .

Example C

Find the n^{th} term rule and thus the 100^{th} term for the arithmetic sequence in which a_3=8 and d=7 .

Solution: This one is a little less straightforward as we will have to first determine the first term from the term we are given. To do this, we will replace a_n with a_3=8 and use 3 for n in the formula to determine the unknown first term as shown:

a_1+(3-1)(7)&=8 \\a_1+2(7)&=8 \\a_1+14&=8 \\a_1&=-6

Now that we have the first term and the common difference we can follow the same process used in the previous example to complete the problem.

a_n&=-6+(n-1)(7) \\&=-6+7n-7 \quad , \text{thus} \ a_n=7n-13. \\&=7n-13

Now we can find the 100^{th} term: a_{100}=7(100)-13=687 .

Intro Problem Revisit From the information given, we can conclude that a_1=1531 and d=76 .

We now have what we need to plug into the rule:

a_n&=1531+(n-1)(76) \\&= 1531+76n-76 \quad , \text{thus the} \ n^{th} \ \text{term rule is} \ a_n=76n+1455

Now to find the 10^{th} term we can use our rule and replace n with 10: a_{10}=76(10) + 1455=760 + 1455 =2215 .

Guided Practice

1. Find the common difference and the n^{th} term rule for the sequence: 5, -3, -11,\ldots

2. Write the n^{th} term rule and find the 45^{th} term for the arithmetic sequence with a_{10}=1 and d=-6 .

3. Find the 62^{nd} term for the arithmetic sequence with a_1=-7 and d=\frac{3}{2} .

Answers

1. The common difference is -3-5=-8 . Now a_n=5+(n-1)(-8)=5-8n+8=-8n+13 .

2. To find the first term:

a_1+(10-1)(-6)&=1 \\a_1-54&=1 \\a_1&=55

Find the n^{th} term rule: a_n=55+(n-1)(-6)=55-6n+6=-6n+61 .

Finally, the 45^{th} term: a_{45}=-6(45)+61=-209 .

3. This time we will not simplify the n^{th} term rule, we will just use the formula to find the 62^{rd} term: a_{62}=-7+(62-1) \left(\frac{3}{2}\right)=-7+61 \left(\frac{3}{2}\right)=- \frac{14}{2}+ \frac{183}{2}= \frac{169}{2} .

Vocabulary

Arithmetic Sequence
A sequence in which the difference between any two consecutive terms is constant.
Common Difference
The value of the constant difference between any two consecutive terms in an arithmetic sequence.

Practice

Identify which of the following sequences is arithmetic. If the sequence is arithmetic find the n^{th} term rule.

  1. 2, 3, 4, 5, \ldots
  2. 6, 2, -1, -3, \ldots
  3. 5, 0, -5, -10, \ldots
  4. 1, 2, 4, 8, \ldots
  5. 0, 3, 6, 9, \ldots
  6. 13, 12, 11, 10, \ldots
  7. 4, -3, 2, -1, \ldots
  8. a, a+2, a+4, a+6, \ldots

Write the n^{th} term rule for each arithmetic sequence with the given term and common difference.

  1. a_1=15 and d=-8
  2. a_1=-10 and d= \frac{1}{2}
  3. a_3=24 and d=-2
  4. a_5=-3 and d=3
  5. a_{10}=-15 and d=-11
  6. a_7=32 and d=7
  7. a_{n-2}=3n+2 , find a_n

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