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.