Recursive and Explicit Formulas for Arithmetic Sequences

​Arithmetic Sequences: Recursive and Explicit Formulas

By Ferdad Roidad

Recursive Formula

Gives the 1st term a(1) and a rule for finding the nth term represented as a(n) given the previous term which is the (n-1)th term represented as a(n-1).

• The recursive formula for a sequence gives the 1st term represented by a(1) and a rule for finding the nth term represented by a(n) given the previous term which is the (n-1)th term represented by a(n-1).

• The drawback is needing all previous terms to find the nth term.

• Any arithmetic sequence can be represented by:

  a(1), a(2), a(3), . . . , a(n-1), a(n), a(n+1), . . .

Recursive Formula

Gives the 1st term a(1) and a rule for finding the nth term represented as a(n) given the previous term which is the (n-1)th term represented as a(n-1).

• Example of recursive formula for the arithmetic sequence -1, 4, 9, 14, 19, . . .

• Recursive formula: a(1) = -1, a(n) = a(n-1) + 5

• The 1st term a(1) is -1 and the common difference d is 5, so we show that the nth term a(n) is equal to the previous term a(n-1) + 5. This means we add 5 to the previous term to get the next term.

• The 6th term a(6) = a(6-1) + 5 = a(5) + 5 = 19 + 5 = 24.

Explicit Formula

Gives the nth term without having to find all previous terms.
a(n) = a(1) + (n-1)(d), where a(1) is the 1st term and d is the common difference found by subtracting any term from the term which follows it.

• The explicit formula allows finding the nth term represented as a(n) without having to find all previous terms.

• The explicit formula can be written as a(n) = a(1) + (n-1)(d), where a(n) is the nth term, a(1) is the 1st term, n is the term number, and d is the common difference for the arithmetic sequence which is either given or can be calculated by subtracting any term from the term which follows it.

• For example, d = a(2) - a(1) = 2nd term - 1st term.

Explicit Formula

​Gives the nth term without having to find all previous terms.
a(n) = a(1) + (n-1)(d), where a(1) is the 1st term and d is the common difference found by subtracting any term from the term which follows it.

• Example of explicit formula for the arithmetic sequence

  -1, 4, 9, 14, 19, . . .

• Explicit formula: a(n) = a(1) + (n-1)(d), so

  a(n) = -1 + (n-1)(5) since a(1) = -1 and d = 4 - (-1) = 5.

• The 6th term a(6) = -1+(6-1)(5) = -1+(5)(5) = -1+25 = 24.

Comparing Recursive and Explicit Formulas for an Arithmetic Sequence

Recursive Formula: a(1) = #, a(n) = a(n-1) + d
Explicit Formula: a(n) = a(1) + (n-1)(d)

• Given the arithmetic sequence: 3, -1, -5, -9, . . .

• We have the 1st term a(1) = 3 and the common difference d = -1 - 3 = -4 which means we add -4 (or subtract 4) to get the next term in the sequence. We also have a(2) = -1, a(3) = -5, a(4) = -9.

• The recursive formula is: a(1) = 3, a(n) = a(n-1) + -4 which can also be written as a(n) = a(n-1) - 4.

• The explicit formula is: a(n) = 3 + (n-1)(-4) which can be simplified using the distributive property to get a(n) = 3 + -4n + 4 which is a(n) = 7 - 4n.

Comparing Recursive and Explicit Formulas for an Arithmetic Sequence

​Recursive Formula: a(1) = #, a(n) = a(n-1) + d
Explicit Formula: a(n) = a(1) + (n-1)(d)

• Given the arithmetic sequence: 3, -1, -5, -9, . . .

• To find the 5th term a(5), we can use either formula since we have a(4) = -9 and the recursive formula would give a(5) = a(5-1) - 4 = a(4) - 4 = -9 - 4 = -13, so a(5) = -13 using the recursive formula.

• The explicit formula gives a(5) = 3 + (5-1)(-4) = 3 + (4)(-4) = 3 + -16 = -13, so a(5) = -13 using the explicit formula.

• We can check the simplified explicit formula a(n) = 7 - 4n as well to show a(5) = 7 - 4(5) = 7 - 20 = -13, so a(5) = -13.

Multiple Choice

Given the recursive formula of an arithmetic sequence, what is the 1st term?

$$a\left(1\right)=-6$$

$$a\left(n\right)=a\left(n-1\right)+4$$

• $$0$$

• $$1$$

• $$-6$$

• $$-2$$

• $$4$$

Multiple Choice

Given the recursive formula of an arithmetic sequence, what is the common difference $$d$$ ?

$$a\left(1\right)=-6$$

$$a\left(n\right)=a\left(n-1\right)+4$$

• $$d=-10$$

• $$d=-2$$

• $$d=-6$$

• $$d=4$$

• $$d=-4$$

Multiple Choice

Given the recursive formula of an arithmetic sequence, what is the 2nd term $$a\left(2\right)$$ ?

$$a\left(1\right)=-6$$

$$a\left(n\right)=a\left(n-1\right)+4$$

• $$a\left(2\right)=-10$$

• $$a\left(2\right)=-2$$

• $$a\left(2\right)=-6$$

• $$a\left(2\right)=5$$

• $$a\left(2\right)=2$$

Multiple Choice

Given the recursive formula of an arithmetic sequence, what is the 3rd term $$a\left(3\right)$$ ?

$$a\left(1\right)=-6$$

$$a\left(n\right)=a\left(n-1\right)+4$$

• $$a\left(3\right)=-14$$

• $$a\left(3\right)=7$$

• $$a\left(3\right)=3$$

• $$a\left(3\right)=2$$

• $$a\left(3\right)=-2$$

Multiple Choice

Given the recursive formula of an arithmetic sequence, what is the explicit formula for this sequence?

$$a\left(1\right)=-6$$

$$a\left(n\right)=a\left(n-1\right)+4$$

Hint: $$a\left(n\right)=a\left(1\right)+\left(n-1\right)\left(d\right)$$ is the explicit formula for an arithmetic sequence.

• a(n) = -6 + (n-1)(-4)

• a(n) = -6 - (n - 1)(4)

• a(n) = -6 + (n - 1)(4)

• a(n) = -6 + (n+1)(4)

• a(n) = 4 + (n - 1)(-6)

Multiple Choice

Given the explicit formula for an arithmetic sequence, what is the 1st term $$a\left(1\right)$$ ?

$$a\left(n\right)=8+\left(n-1\right)\left(3\right)$$

Hint: In general, the explicit formula for an arithmetic sequence is $$a\left(n\right)=a\left(1\right)+\left(n-1\right)\left(d\right)$$

• $$a\left(1\right)=1$$

• $$a\left(1\right)=4$$

• $$a\left(1\right)=8$$

• $$a\left(1\right)=11$$

• $$a\left(1\right)=3$$

Multiple Choice

Given the explicit formula for an arithmetic sequence, what is the common difference $$d$$ ?

$$a\left(n\right)=8+\left(n-1\right)\left(3\right)$$

Hint: In general, the explicit formula for an arithmetic sequence is $$a\left(n\right)=a\left(1\right)+\left(n-1\right)\left(d\right)$$

• $$d=8$$

• $$d=11$$

• $$d=-1$$

• $$d=5$$

• $$d=3$$

Multiple Choice

Given the explicit formula for an arithmetic sequence, what is the 2nd term $$a\left(2\right)$$ ?

$$a\left(n\right)=8+\left(n-1\right)\left(3\right)$$

Hint: Replace $$n$$ with $$2$$ in the given explicit formula, then simplify.

• $$a\left(2\right)=8$$

• $$a\left(2\right)=11$$

• $$a\left(2\right)=27$$

• $$a\left(2\right)=3$$

• $$a\left(2\right)=14$$

Multiple Choice

Given the explicit formula for an arithmetic sequence, what is the $$31st$$ term $$a\left(31\right)$$ ?

$$a\left(n\right)=8+\left(n-1\right)\left(3\right)$$

Hint: Replace $$n$$ with $$31$$ in the given explicit formula, then simplify.

• $$a\left(31\right)=38$$

• $$a\left(31\right)=39$$

• $$a\left(31\right)=41$$

• $$a\left(31\right)=98$$

• $$a\left(31\right)=114$$

Multiple Choice

$$a\left(n\right)=8+\left(n-1\right)\left(3\right)$$ ; $$a\left(101\right)=8+\left(101-1\right)\left(3\right)$$

$$a\left(101\right)=8+\left(100\right)\left(3\right)=\left(108\right)\left(3\right)=324$$

A student found the $$101st$$ term of the arithmetic sequence with the above explicit formula by replacing $$n$$ with $$101$$ .

Select the appropriate choice.

• The student's work and answer are both correct.

• The student should have replaced $$n$$ with $$100$$ .

• The student should have multiplied 100 by 3, then added 8 to get 308.

• The student should have added 8 with 101 to get 109, then subtracted 3 to get 106.

• The student should have found the first 100 terms, then added 8 to the 100th term.

Multiple Choice

Given the explicit formula for an arithmetic sequence is $$a\left(n\right)=-1+\left(n-1\right)\left(6\right)$$ , determine the simplified representation of this explicit formula by applying the distributive property and combining like terms (numbers). Multiply 6 by (n - 1), then combine like terms.

• $$a\left(n\right)=-8n$$

• $$a\left(n\right)=6n-2$$

• $$a\left(n\right)=5n-6$$

• $$a\left(n\right)=6n-7$$

• $$a\left(n\right)=-7n-6$$

Multiple Choice

Given the explicit formula for an arithmetic sequence is $$a\left(n\right)=12+\left(n-1\right)\left(-4\right)$$ , determine the simplified representation of this explicit formula by applying the distributive property and combining like terms (numbers). Multiply -4 by (n - 1), then combine like terms.

• $$a\left(n\right)=7n$$

• $$a\left(n\right)=16-4n$$

• $$a\left(n\right)=12n$$

• $$a\left(n\right)=-48+4n$$

• $$a\left(n\right)=8-4n$$