Question

4. The first term in the sequence is 5 and the fifth term is 17. What is the common difference?

(1) 2.4
(2) 12
(3) 3
(4) 4

Answer 1

(3) 3

Explanation:

Arithmetic Sequence:

An arithmetic sequence is a sequence of numbers in which each of them can be defined by adding a common difference to a term to find the next term.

So for example, let the first term, or number be expressed as:
`a_1` and the common difference be expressed as:
`d` That means the second term, or number can be defined as:
`a_2=a_1+d`. If we now add the common difference to the second term, we would get the third term, and so on.

An arithmetic sequence can be compared to that of a linear equation, except it's a sequence of numbers... and also it's restricted to integers greater than zero (starts at first term, second term.... and so on)

So the common difference can be compared to the slope, and we can actually just use the slope formula! The "x" value being the term number, and the "y" value being the term. So the first term in the sequence is 5 can be expressed as: (1, 5) and the fifth term is 17 can be expressed as (5, 17). Now using the slope formula we would get:
`d=(17-5)/(5-1)=(12)/(4)=3`

Another way to think of it is the first term is 5, and if we add "d" or the common difference to it, we get the second term, if we add another "d" we get the third term, if we another "d" we get the fourth term and if add yet another "d" we get the fifth term. We add a total of four "d's" which can be expressed as:
`5+4d` and this equals the fifth term, but we also know this to be equal to 17, so we can set them equal to each other:
`5+4d=17` now just subtract 5 and divide by 4 to get:
`d=3`