FST Arithmetic Explicit Notes

Arithmetic & Geometric Explicit & Recursive Formulas Notes

Learn how to formulate arithmetic and geometric sequence to find nth term (or how many terms or means)

Terms to Know for Arithmetic Sequences

• Sequence - a list of #s in a particular order

• Term - each # in a sequence

• Arithmetic sequence - a list of #s that each term after first is found by adding constant d

• common difference - difference between 2 successive terms in arithmetic sequence; d = a₂ - a₁

Arithmetic Sequences: Finding the Next Terms

• Find the next 3 terms ; 52, 43, 34

• common difference; d = a₂ - a₁ = 43 - 52 = - 9

• The next 3 terms are 25, 16, and 7

• Find the next 3 terms ; -11, 2, 15

• common difference; d = a₂ - a₁ = 2 - (-11) = 13

• The next 3 terms are 28, 41, and 54

Multiple Choice

Find the next 3 terms in the sequence: 49, 45, 41, 37....

• 33, 29, 25, 21

• 33, 28, 25, 20

• 35, 31, 27, 23

• 33, 29, 26, 22

Arithmetic Sequences: Finding the n^th term

• a_n means n^th term, a₁ means the 1^st term, and a₀ means the term before the 1st term; n indicate a position. (such as 1st, 2nd, 3rd, or 100th)

• a_n = d⋅n + a_0; leave n as "n" when you write explicit formula.

• 11,13, 15, 17, 19

• d = 2, a₁ = 11, so a₀ = 9;

• a_n = 2n + 9

Multiple Choice

Find the nth term of...
8,11,14,17... a_n = dn + a₀

• a_n = 5n+3

• a_n = 5n-2

• a_n = 3n+5

• a_n = -2n+5

Multiple Choice

Find the nth term of...
10,20,30,40... a_n = dn + a₀

• a_n = 3n

• a_n = 2n

• a_n = 4n

• a_n = 10n

Fill in the Blanks

What is the simplified equation for the nth term of the arithemtic sequence:

18, 12, 6, 0, -6, ...

(Express your answer in A(n) = dn + a₀ where d and a₀ are integers)

Arithmetic Recursive Formula;

a₁ = #, a_n = a_n-1 + d

• A rule in which one or more previous terms are used to generate the next term; a_n-1 means a previous term; a₁ = #, a_n = a_n-1 + d

• The Fibonnaci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . ;a_n = a_n-1 + a_n-2

• 1) a₁ = 14; a_n = a_n-1 + 9

• Find the 1st 5 terms ; 14, 23, 32, 41, 50

• 2) a₁ = 6; a_n = a_n-1 - 5

• The first 5 terms are 6, 1, -4, - 9, - 14

Multiple Choice

3) Find the 1st 5 terms; a₁ = 354, a_n = a_n-1 + 6

• 354, 360, 366, 372, 378

• 360, 366, 372, 378, 384

• 366, 372, 378, 384, 390

• 372, 378, 384, 390, 396

• 354, 348, 342, 336, 330

Multiple Choice

4) Given the recursive formula below, determine a₄.

a₁ = 5

a_n = a_n-1 + 5

• 25

• 5

• 10

• 20

Multiple Choice

5) Write a recursive rule for each sequence. Then give the next 3 terms. {4, 11, 18, 25, 32, . . . }

• a₁ = 4,

  a_n = a_n-1 - 7;

  41, 50, 59

• a₁ = 4,

  a_n = a_n-1 + 7;

  39, 46, 53

• a₁ = 4,

  a_n = 7 a_n-1,

  _{123, 145, 189}

• a₁ = 4,

  a_n = a_n-1 - 6;

  38, 44, 50

• a₁ = 4,

  a_n = a_n-1 + 6;

  38, 44, 50

Terms to Know for Geometric Sequences

• Geometric sequence - a list of #s with multiplying a constant "r" pattern; 3, 6, 12, 24,...

• Common ratio; r = 2nd term/1st term

Multiple Select

1) Determine whether the sequence is geometric. If yes, identify the common ratio and give the next 3 terms.

{6, 12, 24, 48, . . . }

• Yes, geometric; r = 2

• 96, 192, 384

• No, arithmetic; d = 6

• Neither arithmetic nor geometric

• 56, 64, 72

Multiple Select

3) Determine whether the sequence is geometric. If yes, identify the common ratio and give the next 3 terms.

{-2, 10, -50, 250, . . . }

• Yes, geometric; r = - 5

• -1250, 6250, -31250

• Yes, geometric; r = 5

• Neither arithmetic nor geometric

• 1250, 6250, 31250

Multiple Select

4) Determine whether the sequence is geometric. If yes, identify the common ratio and give the next 3 terms.

{4, 8, 20, 60, . . . }

• Yes, geometric; r = - 5

• Yes, geometric; r = 5

• Neither arithmetic nor geometric

• No, it is arithmetic

Multiple Choice

8) Write the explicit formula for the following geometric sequence:

100, 50, 25, 12.5, ...

^a_n= a₁(r)^n-1

• a_n = 2(100)^n-1

• a_n = 100(2)^n-1

• a_n = 100(1/2)^n-1

• a_n = (1/2)(100)^n-1

Geometric Recursive Formula; uses previous term.

a₁ = #, a_n = r(a_n-1); r = a₂ ÷ a₁

• Write a recursive rule, then find a_5. {1, 5, 25, 125, . . . }

• a₁ = 1; a_n = 5(a_n-1); a₅ = 625

• Write a recursive rule, then find a_5. {130, 65, 32.5, 16.25, . . . }

• a₁ = 130; a_n = 0.5(a_n-1); a₅ = 8.125

Multiple Select

5) Write a recursive rule for each sequence. Then find a₅. {2, - 8, 32, -128, . . . }

• a₁ = 4, a_n = a_n-1 - 7

• a₁ = 2, a_n = - 4( a_n-1)

• a₁ = 2, a_n = 4 a_n-1

• a₅ = 512

• a₅ = - 512

Multiple Select

6) Write a recursive rule for each sequence. Then find a₅. {8, 24, 72, 216, . . .

• a₁ = 8, a_n = 3(a_n-1)

• a₁ = 8, a_n = - 3( a_n-1)

• a₁ = 8, a_n = 4 a_n-1

• a₅ = 648

• a₅ = - 648

Geometric Explicit: Finding the n^th term;

a_n = a₁⋅r ^n-1

• Explicit Formula: A rule in which the nth term is defined as a function of n (Previous terms are unnecessary). We use it to find nth term or how many terms are there.

• a_n = a₁⋅r ^n-1_; leave n - 1as "n - 1" for explicit.

• 15) a₁ = 7 and r = 4; find a₈.

• Use a_n = a₁⋅r ^n-1

• a₈ = 7(4)^8-1= 7(4)⁷ = 114688

Multiple Choice

7) Write the explicit formula for the sequence given by 3, 6, 12, 24,

^a_n= a₁(r)^n-1

• ^a_n= 3(2)^n-1

• ^a_n= 3+(2)ⁿ

• ^a_n= 3(2)ⁿ

• ^a_n= 3+(2)^n-1

Multiple Choice

Create the arithmetic explicit formula for the sequence:

-8, -6, -4, ...

(Hint: Write your formula and then simplify it.)

a_n = dn + a₀

• a_n= - 2+10n

• a_n= -6 + 2n

• a_n= 2n - 10

• a_n= - 6 - 2n

Multiple Choice

Find the 26th term in the sequence: 20, 26, 32, 38, ...

• 182
• 176
• 170
• -118

Multiple Choice

Find a₁₂ for the geometric sequence defined by the following formula:

a_n = 6(2)^n-1

• 4096

• 24,576

• 2048

• 12,288

Multiple Choice

Create the arithmetic explicit formula for the sequence:

23, 7, - 9, ...

(Hint: Write your formula and then simplify it.)

a_n = dn + a₀

• a_n= - 16n + 39

• a_n= -16 + 33n

• a_n= 39n - 16

• a_n= - 16 - 39n