Question

Describe and explain the difference between explicit formulas for arithmetic

sequences and explicit formulas for geometric sequences.

Answer 1

The difference between explicit formulas for arithmetic and geometric sequences lies in the common difference or ratio. The explicit formula for arithmetic sequences is An = a1 + (n-1)d, while the explicit formula for geometric sequences is An = a1 * r^(n-1).

Step-by-step explanation:

The difference between explicit formulas for arithmetic and geometric sequences lies in the nature of the sequence itself.

Arithmetic sequences have a common difference between consecutive terms, while geometric sequences have a common ratio between consecutive terms.

To find an explicit formula for an arithmetic sequence, you can use the formula:

An = a1 + (n-1)d

where An represents the nth term, a1 is the first term, and d is the common difference.

For geometric sequences, the explicit formula is given by:

An = a1 * r^(n-1)

where An represents the nth term, a1 is the first term, and r is the common ratio.

Answer 2

Arithmetic sequence:
`a_(n)=a_(1)+(n-1)d`

Geometric sequence:
`a_(n)=a_(1)(r)^(n-1)`

Step-by-step explanation:

The arithmetic sequence: is the sequence whos terms increased or decreased by a constant amount.

Examples:

• 4, 7, 10, 13, 16, ........................ (increased by 3)

• 25, 20, 15, 10, ......................... (Decreased by 5)

The explicit formula for the nth term of the arithmetic sequence is:

• 
  `a_(n)=a_(1)+(n-1)d`

• 
  `a_(1)` is the first term

• d is the constant difference between each two consecutive terms

• n is the position of the number in the sequence

The geometric sequence: is the sequence whos consecutive terms have a constant ratio

Examples:

• 1, 2, 4, 8, 16, ........................ (Multiplying by 2)

• 625, 125, 25, 5, ......................... (Dividing by 5)

The explicit formula for the nth term of the geometric sequence is:

• 
  `a_(n)=a_(1)(r)^(n-1)`

• 
  `a_(1)` is the first term

• r is the constant ratio between each two consecutive terms

• n is the position of the number in the sequence

* Arithmetic sequence →→→→→→ Geometric sequence

Has a constant difference →→→→→→ Has a constant ratio

`a_(n)=a_(1)+(n-1)d` →→→→→→
`a_(n)=a_(1)(r)^(n-1)`

`d=a_(n)-a_(n-1)` →→→→→→
`r=(a_(n))/(a_(n-1) )`