Arithmetic Sequences
They are lists of numbers with a predictable pattern: Each term is obtained as the previous one plus a constant number, called common difference.
The sequence given in the question is: -3, 4, 11, 18,...
The common difference can be obtained subtracting two consecutive terms, for example:
r = 4 - (-3) = 7
We can also get the same number subtracting the 3rd term with the 2nd term:
r = 11 - 4 = 7
Every time we do that, we get the very same number, thus we know the common difference is r = 7.
Another important parameter is the first term, in our case it is
a1 = -3
The number besides the 'a' means the counter of terms, thus the second term is called a2 and can be computed as:
a2 = a1 + r = -3 + 7 = 4
The third term is calculated by:
a3 = a2 + r = 4 + 7 = 11
But it also can be obtained by
a3 = a1 + 2*7 = -3 + 14 = 11
Inferring the general formula for an arithmetic sequence we get
an = a1 + (n-1)*r
a)
If we wanted to calculate the term 58, the formula will become:
a58 = -3 + 7 * (58-1)
Since Amanda used the formula
a58 = 18 + 7 * (58-1)
Her mistake was the incorrect value of a1.
b)
The correct value is
a58 = -3 + 7 * 57 = -3 + 399 = 396
a58 = 396