Question

arithmetic sequences practice # 1

1. is the sequence arithmetic or not?

{8, 16, 32, 64...}

1. find the next 3 terms in the sequence.

{-5, -1, 3, 7, 11...}

1. is the sequence finite or infinite?

{14, 9, 4, -1, -6...}

1. what is the common difference in this sequence?

{82, 18, -46, -110}

1. find the next 5 terms in the sequence.

{16, 25, 34...}

1. the first term in a sequence is 15. the common difference is -4. write the first 5 terms of the sequence.
2. write an explicit rule for the nth term of the sequence: {3, 16, 29...}. then find a₂₄.
3. write an explicit rule for the nth term of the sequence: {94, 67, 40...}. then find a₇₅.

Explanation:

Step1: Recall arithmetic - sequence properties

An arithmetic sequence has a common difference \(d=a_{n + 1}-a_{n}\).

Step2: Solve problem 2

The common difference \(d = 4\) for the sequence \(\{-5,-1,3,7,11,\cdots\}\).
The next 3 terms:
\(a_{6}=11 + 4=15\)
\(a_{7}=15 + 4 = 19\)
\(a_{8}=19+4 = 23\)

Step3: Solve problem 4

For the sequence \(\{82,18,-46,-110,\cdots\}\), \(d=18 - 82=-64\)

Step5: Solve problem 5

The common difference \(d = 9\) for the sequence \(\{16,25,34,\cdots\}\)
The next 5 terms:
\(a_{4}=34 + 9=43\)
\(a_{5}=43 + 9 = 52\)
\(a_{6}=52+9 = 61\)
\(a_{7}=61 + 9=70\)
\(a_{8}=70+9 = 79\)

Step6: Solve problem 6

The first - term \(a_{1}=15\) and \(d=-4\)
The first 5 terms:
\(a_{1}=15\)
\(a_{2}=15-4 = 11\)
\(a_{3}=11-4 = 7\)
\(a_{4}=7-4 = 3\)
\(a_{5}=3-4=-1\)

Step7: Solve problem 8

For the sequence \(\{3,16,29,\cdots\}\), \(a_{1}=3\) and \(d = 13\)
The explicit rule \(a_{n}=a_{1}+(n - 1)d=3+(n - 1)\times13=3+13n-13=13n - 10\)
\(a_{24}=13\times24-10=312 - 10=302\)

Step8: Solve problem 9

For the sequence \(\{94,67,40,\cdots\}\), \(a_{1}=94\) and \(d=-27\)
The explicit rule \(a_{n}=a_{1}+(n - 1)d=94+(n - 1)\times(-27)=94-27n + 27=121-27n\)
\(a_{75}=121-27\times75=121 - 2025=-1904\)

Answer:

1. \(15,19,23\)
2. \(-64\)
3. \(43,52,61,70,79\)
4. \(15,11,7,3,-1\)
5. Explicit rule: \(a_{n}=13n - 10\), \(a_{24}=302\)
6. Explicit rule: \(a_{n}=121-27n\), \(a_{75}=-1904\)