Question

How do you find the explicit formula for the following sequence 20, 15, 10, 5, 0?

Answer 1

As we can see that the given sequence is an arithmetic sequence i.e., the terms obtained is by adding or subtracting a constant to the preceding term, there is a constant difference between consecutive terms and hence by this, we can find the formula for the sequence.

Formula used:
${a_n} = {a_1} + \left( {n - 1} \right)d$
${a_n}$is the nth term
${a_1}$ is the first term
$n$ is the terms number in the sequence
$d$ is the common difference

Complete step by step solution:
This is an arithmetic sequence since there is a common difference between each term. In this case, subtracting 5 to the previous term in the sequence gives the next term.
An explicit formula of an arithmetic series includes all given information as
$\Rightarrow {a_n} = {a_1} + \left( {n - 1} \right)d$
This is the formula of an arithmetic sequence.
In the given arithmetic sequence, we get d as
$\Rightarrow d = {a_2} - {a_1}$
$\Rightarrow d = 15 - 20 = - 5$
Substitute in the values of a1 = 20 and d = -5 in
$\Rightarrow {a_n} = {a_1} + \left( {n - 1} \right)d$
$\Rightarrow {a_n} = 20 + \left( {n - 1} \right)\left( { - 5} \right)$
To simplify each term, apply distributive property
$\Rightarrow {a_n} = 20 - 5n - 5\left( { - 1} \right)$
Multiplying the terms, we get
$\Rightarrow {a_n} = 20 - 5n + 5$
Therefore, we get
$\Rightarrow {a_n} = 25 - 5n$
Hence, by applying this formula we can get the following sequence.

Additional information:
An arithmetic progression (AP) is a sequence of numbers in which each succeeding number is obtained either by adding or subtracting a specific number called common difference. The general form of AP is: a, a + d, a + 2d, ….
A sequence of numbers is said to be a harmonic progression if the reciprocal of those numbers is in AP. Suppose A, G, H are the means of AP, GP and HP, respectively, then: ${G^2} = AH$

Note: In an arithmetic sequence, the terms can be obtained by adding or subtracting a constant to the preceding term, there is a constant difference between consecutive terms, the sequence is said to be an arithmetic sequence, wherein in case of geometric progression each term is obtained by multiplying or dividing a constant to the preceding term.