If the first term of an arithmetic sequence is 10 and the 13th term is 58, what is the common difference?

Find the 53rd term of the arithmetic sequence -12,-1,10

The nth term of an arithmetic progression is defined by

n = a + (n-1)d where,
n = the nth term
a = the first term
d = the common difference

Therefore,
58 = 10 + (13 - 1)d
48 = 12d
d = 4

To find the common difference of an arithmetic sequence, we can use the formula:

\[a_n = a_1 + (n-1)d\]

Where:
- \(a_n\) represents the \(n\)th term of the sequence
- \(a_1\) represents the first term of the sequence
- \(n\) represents the position of the term in the sequence
- \(d\) represents the common difference

In this case, we know that the first term, \(a_1\), is 10 and the 13th term, \(a_{13}\), is 58. We can plug these values into the formula to solve for the common difference (\(d\)):

\[58 = 10 + (13-1)d\]

Simplifying the equation:

\[58 = 10 + 12d\]

Subtracting 10 from both sides:

\[48 = 12d\]

Dividing both sides by 12:

\[4 = d\]

Therefore, the common difference of the arithmetic sequence is 4.