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Converting recursive & explicit forms of arithmetic sequences

Learn how to convert between recursive and explicit formulas of arithmetic sequences.
Before taking this lesson, make sure you know how to find recursive formulas and explicit formulas of arithmetic sequences.

Converting from a recursive formula to an explicit formula

An arithmetic sequence has the following recursive formula.
{a(1)=3a(n)=a(n1)+2\begin{cases} a(1)=\greenE 3 \\\\ a(n)=a(n-1)\maroonC{+2} \end{cases}
Recall that this formula gives us the following two pieces of information:
  • The first term is start color #0d923f, 3, end color #0d923f
  • To get any term from its previous term, add start color #ed5fa6, 2, end color #ed5fa6. In other words, the common difference is start color #ed5fa6, 2, end color #ed5fa6.
Let's find an explicit formula for the sequence.
Remember that we can represent a sequence whose first term is start color #0d923f, A, end color #0d923f and common difference is start color #ed5fa6, B, end color #ed5fa6 with the standard explicit form start color #0d923f, A, end color #0d923f, plus, start color #ed5fa6, B, end color #ed5fa6, left parenthesis, n, minus, 1, right parenthesis.
Therefore, an explicit formula of the sequence is a, left parenthesis, n, right parenthesis, equals, start color #0d923f, 3, end color #0d923f, start color #ed5fa6, plus, 2, end color #ed5fa6, left parenthesis, n, minus, 1, right parenthesis.

Check your understanding

1) Write an explicit formula for the sequence.
{b(1)=22b(n)=b(n1)+7\begin{cases} b(1)=-22 \\\\ b(n)=b(n-1)+7 \end{cases}
b, left parenthesis, n, right parenthesis, equals

2) Write an explicit formula for the sequence.
{c(1)=8c(n)=c(n1)13\begin{cases} c(1)=8 \\\\ c(n)=c(n-1)-13 \end{cases}
c, left parenthesis, n, right parenthesis, equals

Converting from an explicit formula to a recursive formula

Example 1: Formula is given in standard form

We are given the following explicit formula of an arithmetic sequence.
d, left parenthesis, n, right parenthesis, equals, start color #0d923f, 5, end color #0d923f, start color #ed5fa6, plus, 16, end color #ed5fa6, left parenthesis, n, minus, 1, right parenthesis
This formula is given in the standard explicit form start color #0d923f, A, end color #0d923f, plus, start color #ed5fa6, B, end color #ed5fa6, left parenthesis, n, minus, 1, right parenthesis where start color #0d923f, A, end color #0d923f is the first term and that start color #ed5fa6, B, end color #ed5fa6 is the common difference. Therefore,
  • the first term of the sequence is start color #0d923f, 5, end color #0d923f, and
  • the common difference is start color #ed5fa6, 16, end color #ed5fa6.
Let's find a recursive formula for the sequence. Recall that the recursive formula gives us two pieces of information:
  1. The first term left parenthesiswhich we know is start color #0d923f, 5, end color #0d923f, right parenthesis
  2. The pattern rule to get any term from the term that comes before it left parenthesiswhich we know is "add start color #ed5fa6, 16, end color #ed5fa6"right parenthesis
Therefore, this is a recursive formula for the sequence.
{d(1)=5d(n)=d(n1)+16\begin{cases} d(1)=\greenE 5\\\\ d(n)=d(n-1)\maroonC{+16} \end{cases}

Example 2: Formula is given in simplified form

We are given the following explicit formula of an arithmetic sequence.
e, left parenthesis, n, right parenthesis, equals, 10, plus, 2, n
Note that this formula is not given in the standard explicit form start color #0d923f, A, end color #0d923f, plus, start color #ed5fa6, B, end color #ed5fa6, left parenthesis, n, minus, 1, right parenthesis.
For this reason, we can't simply use the structure of the formula to find the first term and the common difference. Instead, we can find the first two terms:
  • e, left parenthesis, start color #11accd, 1, end color #11accd, right parenthesis, equals, 10, plus, 2, dot, start color #11accd, 1, end color #11accd, equals, 12
  • e, left parenthesis, start color #11accd, 2, end color #11accd, right parenthesis, equals, 10, plus, 2, dot, start color #11accd, 2, end color #11accd, equals, 14
Now we can see that the first term is start color #0d923f, 12, end color #0d923f and the common difference is start color #ed5fa6, 2, end color #ed5fa6.
Therefore, this is a recursive formula for the sequence.
{e(1)=12e(n)=e(n1)+2\begin{cases} e(1)=\greenE{12}\\\\ e(n)=e(n-1)\maroonC{+2} \end{cases}

Check your understanding

3) The explicit formula of an arithmetic sequence is f, left parenthesis, n, right parenthesis, equals, 5, plus, 12, left parenthesis, n, minus, 1, right parenthesis.
Complete the missing values in the recursive formula of the sequence.
{f(1)=Af(n)=f(n1)+B\begin{cases} f(1)=A\\\\ f(n)=f(n-1)+B \end{cases}
A, 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
B, 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

4) The explicit formula of an arithmetic sequence is g, left parenthesis, n, right parenthesis, equals, minus, 11, minus, 8, left parenthesis, n, minus, 1, right parenthesis.
Complete the missing values in the recursive formula of the sequence.
{g(1)=Ag(n)=g(n1)+B\begin{cases} g(1)=A\\\\ g(n)=g(n-1)+B \end{cases}
A, 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
B, 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

5) The explicit formula of an arithmetic sequence is h, left parenthesis, n, right parenthesis, equals, 1, plus, 4, n.
Complete the missing values in the recursive formula of the sequence.
{h(1)=Ah(n)=h(n1)+B\begin{cases} h(1)=A\\\\ h(n)=h(n-1)+B \end{cases}
A, 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
B, 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

6) The explicit formula of an arithmetic sequence is i, left parenthesis, n, right parenthesis, equals, 23, minus, 6, n.
Complete the missing values in the recursive formula of the sequence.
{i(1)=Ai(n)=i(n1)+B\begin{cases} i(1)=A\\\\ i(n)=i(n-1)+B \end{cases}
A, 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
B, 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

Challenge problem

7*) Select all the formulas that correctly represent the arithmetic sequence 101, comma, 114, comma, 127, comma, point, point, point
Choose all answers that apply:
Choose all answers that apply:

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