Question

Determine the common difference, the fifth term, and the sum of the first 100 terms of the following sequence: 1, 2.5, 4, 5.5

Answer 1

Given:

sequence: 1, 2.5, 4, 5.5, ...

To determine the common difference, the fifth term, and the sum of the first 100 terms of the given sequence, we first note the formula of the arithmetic sequence as shown below:

`\begin{gathered} a_n=a_1+(n-1)d \\ where: \\ a_1=first\text{ term} \\ n=nth\text{ term} \\ d=common\text{ difference} \end{gathered}`

We also note that:

`\begin{gathered} a_1=1 \\ a_2=2.5 \\ a_3=4 \\ a_4=5.5 \end{gathered}`

Next, we find the common diffence by:

`\begin{gathered} a_2-a_1=2.5-1=1.5 \\ a_3-a_2=4-2.5=1.5 \\ a_4-a_3=5.5-4=1.5 \end{gathered}`

Hence, the common difference is: 1.5

Then, we plug in a1=1, n=5 and d=1.5 into the formula of the arithmetic sequence to find the fifth term:

`\begin{gathered} a_(n)=a_(1)+(n-1)d \\ a_5=1+(5-1)(1.5) \\ Simplify \\ a_5=1+(4)(1.5) \\ a_5=7 \end{gathered}`

Hence, the fifth term is: 7

Now, we get the sum of the first 100 terms by using the formula:

`\begin{gathered} S=(n)/(2)(2a_1+(n-1)d) \\ where: \\ S=Sum \\ a_1=1 \\ n=100 \\ d=1.5 \end{gathered}`

We plug in what we know:

`\begin{gathered} S=(n)/(2)(2a_(1)+(n-1)d) \\ S=(100)/(2)(2(1)+(100-1)(1.5)) \\ Simplify \\ S=50(150.5) \\ S=7525 \end{gathered}`

Hence, the sum of the first 100 terms is: 7525

Therefore, the answers are:

Common difference: 1.5

Fifth term: 7

Sum of the first 100 terms: 7525