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Arithmetic sequences review

Review arithmetic sequences and solve various problems involving them.

Parts and formulas of arithmetic sequences

In arithmetic sequences, the difference between consecutive terms is always the same. We call that difference the common difference.
For example, the common difference of the following sequence is plus, 2:
start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6
3, comma5, comma7, comma9, comma, point, point, point
Arithmetic sequence formulas give a, left parenthesis, n, right parenthesis, the n, start superscript, start text, t, h, end text, end superscript term of the sequence.
This is the explicit formula for the arithmetic sequence whose first term is start color #11accd, k, end color #11accd and common difference is start color #ed5fa6, d, end color #ed5fa6:
a, left parenthesis, n, right parenthesis, equals, start color #11accd, k, end color #11accd, plus, left parenthesis, n, minus, 1, right parenthesis, start color #ed5fa6, d, end color #ed5fa6
This is the recursive formula of that sequence:
{a(1)=ka(n)=a(n1)+d\begin{cases}a(1) = \blueD k \\\\ a(n) = a(n-1)+\maroonC d \end{cases}
Want to learn more about arithmetic sequences? Check out this video.

Extending arithmetic sequences

Suppose we want to extend the sequence 3, comma, 8, comma, 13, comma, point, point, point We can see each term is start color #ed5fa6, plus, 5, end color #ed5fa6 from the previous term:
start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6
3, comma8, comma13, comma, point, point, point
So we simply add that difference to find that the next term is 18:
start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6
3, comma8, comma13, comma18, comma, point, point, point
Problem 1
What is the next term in the sequence minus, 5, comma, minus, 1, comma, 3, comma, 7, comma, dots?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Want to try more problems like this? Check out this exercise.

Writing recursive formulas

Suppose we want to write a recursive formula for 3, comma, 8, comma, 13, comma, point, point, point We already know the common difference is start color #ed5fa6, plus, 5, end color #ed5fa6. We can also see that the first term is start color #11accd, 3, end color #11accd. Therefore, this is a recursive formula for the sequence:
{a(1)=3a(n)=a(n1)+5\begin{cases}a(1) = \blueD 3 \\\\ a(n) = a(n-1)\maroonC{+5} \end{cases}
Problem 1
Find k and d in this recursive formula of the sequence minus, 5, comma, minus, 1, comma, 3, comma, 7, comma, dots.
{a(1)=ka(n)=a(n1)+d\begin{cases}a(1) = k \\\\ a(n) = a(n-1)+d \end{cases}
k, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
d, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Want to try more problems like this? Check out this exercise.

Writing explicit formulas

Suppose we want to write an explicit formula for 3, comma, 8, comma, 13, comma, point, point, point We already know the common difference is start color #ed5fa6, plus, 5, end color #ed5fa6 and the first term is start color #11accd, 3, end color #11accd. Therefore, this is an explicit formula for the sequence:
a, left parenthesis, n, right parenthesis, equals, start color #11accd, 3, end color #11accd, start color #ed5fa6, plus, 5, end color #ed5fa6, left parenthesis, n, minus, 1, right parenthesis
Problem 1
Write an explicit formula for minus, 5, comma, minus, 1, comma, 3, comma, 7, comma, dots
a, left parenthesis, n, right parenthesis, equals

Want to try more problems like this? Check out this exercise.

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