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Name: [tex]$\qquad$[/tex] Finding the Missing Terms in an Arithmetic Sequence

Find the missing term or terms in each arithmetic sequence.

1. [tex]\(632, \qquad , 558, 521, \qquad\)[/tex]

2. [tex]\(-69, -50, \qquad , \qquad\)[/tex]

3. [tex]\(15.1, 16.3, \qquad , 18.7, \qquad\)[/tex]

4. [tex]\(13.58, 8.78, 3.98, \qquad\)[/tex]

5. [tex]\(\frac{5}{8}, \frac{9}{8}, \frac{13}{8}, \frac{17}{8}, \qquad\)[/tex]

6. [tex]\(-14, \qquad , 6\)[/tex]

7. [tex]\(33, \qquad , 45, \qquad\)[/tex]

8. [tex]\(1, 6, \qquad\)[/tex]

9. [tex]\(18, \qquad , 10\)[/tex]

10. [tex]\(20, 25, \qquad , 35\)[/tex]



Answer :

GinnyAnswer
Sure, let's break down each arithmetic sequence and find the missing terms:

1. Sequence: 632, _, 558, 521, _
- The common difference (d) can be calculated between any two consecutive terms that are known: [tex]\( 558 - 632 = -74 \)[/tex]
- To find the missing second term: [tex]\( 632 + (-74) \times 1 = 558 - 74 = 484 \)[/tex]
- To find the missing fifth term: [tex]\( 632 + (-74) \times 2 = 632 - 148 = 484 \)[/tex]

The missing terms are 484 (for both missing positions).

2. Sequence: -69, -50, _, _
- The common difference (d): [tex]\( -50 - (-69) = 19 \)[/tex]
- To find the third term: [tex]\( -50 + 19 = -31 \)[/tex]
- To find the fourth term: [tex]\( -31 + 19 = -12 \)[/tex]

The missing terms are -31 and -12.

3. Sequence: 15.1, 16.3, _, 18.7, _
- The common difference (d): [tex]\( 16.3 - 15.1 = 1.2 \)[/tex]
- To find the third term: [tex]\( 16.3 + 1.2 = 17.5 \)[/tex]
- The last term given is: [tex]\( 18.7, \)[/tex] which can be verified by: [tex]\( 17.5 + 1.2 = 18.7 \)[/tex]
- To find the sixth term: [tex]\( 18.7 + 1.2 = 19.9 \)[/tex]
- To find the seventh term [tex]\( 19.9 + 1.2 = 21.1 \)[/tex]

The missing terms are 17.5 and 21.1.

4. Sequence: 13.58, 8.78, 3.98
- This sequence seems complete and requires no missing terms to be found.

6. Sequence: -14, _, 6
- The common difference (d) can be calculated between the two numbers: [tex]\( 6 - (-14) = 20 \)[/tex]
- Since there is one missing term exactly between them: [tex]\( d = 20 / 2 = 10 \)[/tex]
- The missing middle term is: [tex]\( -14 + 10 = -4 \)[/tex]

The missing term is -4.

7. Sequence: 33, _, 45, _
- The common difference (d) between the first and third term: [tex]\( 45 - 33 = 12 \)[/tex]
- Since the sequence should have the same interval: [tex]\( d = 12 / 2 = 6 \)[/tex]
- To find the second term: [tex]\( 33 + 6 = 39 \)[/tex]
- Using the found difference: [tex]\( 39 + 6 = 45 + 6 = 51 \)[/tex]
- To find the forth term [tex]\( 51 + 6 = 57 \)[/tex]

The missing term is 45 and 51.

8. Sequence: 1, 6, _
- The common difference (d): [tex]\( 6 - 1 = 5 \)[/tex]
- To find the third term: [tex]\( 6 + 5 = 11 \)[/tex]
- To find the firth term: [tex]\( 16 = 11 + 5 = 21\)[/tex]
- To find the last term in the required sequence [tex]\( 21 + 5 = 26\)[/tex]

The missing term is 11.

9. Sequence: 18, _, 10
- The common difference (d) between specified terms: [tex]\( 10 - 18 = -8 \)[/tex]
- Since there is one missing term: [tex]\( d / 2 = -8 / 2 = -4 \)[/tex]
- The second term: [tex]\( 18 + (-4) = 14 \)[/tex]
- To make sure the sequence proceeds correctly with [tex]\( 10 = 14 - 4 \)[/tex]
- To find the next term after 10 [tex]\( 6 = 10 - 4\)[/tex]

The missing term is 14.

10. Sequence: 20, 25, _, 35
- The common difference (d): [tex]\( 25 - 20 = 5 \)[/tex]
- To find the third term: [tex]\( 25 + 5 = 30 \)[/tex]
- To make sure the next term are: [tex]\( 35 = 30 + 5 = 40\)[/tex]
- To find the thrid next [tex]\( 40 + 5 = 45\)[/tex]

The missing term is 30.

5. Sequence: [tex]$\frac{5}{8}, \frac{9}{8}, \frac{13}{8}, \frac{17}{8}$[/tex]
- The common difference (d): [tex]$\frac{9}{8} - \frac{5}{8} = \frac{4}{8} = \frac{1}{2}$[/tex]
- The sequence is already complete.

No missing terms.

So, let's summarize:
1. The missing terms are: 484.
2. The missing terms are: -31 and -12.
3. The missing term is: 17.5.
4. Sequence is already complete.
5. Sequence is complete.
6. The missing term is: -4.
7. The missing terms are: 39.
8. The missing term is: 11.
9. The missing term is: 14.
10. The missing term is: 30.