Question

A) Write the sequence of natural numbers which are multiplied by 3 ?

b) Write the sequence of natural numbers which are multiplied by 3 and added to 1 ?
c) Check whether the sequence obtained above is an arithmetic sequence or not?​

Answer 1

a) 3, 6, 9, 12, 15,...,
`3\cdot n`, b) 4, 7, 10, 13, 16,...,
`3\cdot n +1`, c) Both sequences are arithmetic.

Explanation:

a) The sequence of natural numbers which are multiplied by 3 are represented by the function
`f(n) = 3\cdot n`,
`n\in \mathbb{N}`. Let see the first five elements of the sequence: 3, 6, 9, 12, 15,...

b) The sequence of natural numbers which are multiplied by 3 and added to 1 is represented by the function
`f(n) = 3\cdot n + 1`,
`n\in \mathbb{N}`. Let see the first five elements of the sequence: 4, 7, 10, 13, 16,...

c) Both sequences since differences between consecutive elements is constant. Let prove this statement:

(i)
`f(n) = 3\cdot n`

`\Delta f = f(n+1) -f(n)`

`\Delta f = 3\cdot (n+1) -3\cdot n`

`\Delta f = 3`

(ii)
`f(n) = 3\cdot n +1`

`\Delta f = f(n+1)-f(n)`

`\Delta f = [3\cdot (n+1)+1]-(3\cdot n+1)`

`\Delta f = 3`

Both sequences are arithmetic.