Question

Describe and explain the difference between explicit formulas for arithmetic
sequences and explicit formulas for geometric sequences.

Answer 1

Answer:

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

 $a_{n}=a_{1}+(n-1)d$                  →→→→→→    $a_{n}=a_{1}(r)^{n-1}$

 $d=a_{n}-a_{n-1}$                          →→→→→→    $r=\frac{a_{n}}{a_{n-1} }$