arithmetic series and applications

Fomula for finding the \(n^{th}\) term of arithmetic sequence: \(a_n=a_1+(n−1)d\) (\(a_1\) is the first term, \(n\) is the term you want to find (eg for finding the \(7^{th}\) term, \(n\) would be \(7\)), and \(d\) is the common difference).

Formula for finding the sum of an arithmetic sequence: \(S_n=\dfrac{n(a_1 + a_n)}{2}\) (\(a_1\) is the first term, \(n\) is the number of terms, and \(a_n\) is the last term).

1)

So we find the last term with the first formula:

\(10 + (8-1)2\) \(=24\).

And the sum with the second formula:

\(\dfrac{8(10+24)}{2}=136\)

So the number of miles Mercedes will ride over the course of \(8\) weeks is \(\boxed{136}\)

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2)

Likewise we can solve this one with the two formulas.

We find the \(7^{th}\) term with the first formula:

\(15+(7-1)3\) \(= 33\)

And the sum with the second formula:

\(\dfrac{7(15+33)}{2}=168\)

So the total number of logs in the stack is \(\boxed{168}\)

\(Q.E.D\)

Number of term =7
First term =15
Common difference =3
Sum =N/2 * [2*F + (N - 1) * D], where N=Number of terms, F=First term, D = Common difference.
Sum = 7/2 * [2*15 + (7 - 1) * 3]
         = 3.5 * [30    +  (6 * 3)    ]
         = 3.5 * [30 + 18]
         = 3.5 * 48
         =168 - Total number of logs in the stack.