Question

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

Answer 1

Answer:

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

Step-by-step 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.