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.