What is the 100th term of the arithmetic sequence 3/4, 5.25, 39/4, 14.25,...? Express answer as a decimal to the nearest hundredth.
difference = 18/4
3/4 + 99(18/4) = 102/4 = 25.50
Thanks for the help!
To find the 100th term of an arithmetic sequence, we need to use the formula for the nth term of an arithmetic sequence:
\[ a_n = a_1 + (n-1)d \]
where \(a_n\) represents the nth term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference between consecutive terms.
In this case, the first term \(a_1\) is 3/4, and we need to find the common difference \(d\).
To find the common difference, we can subtract any two consecutive terms in the sequence. Let's subtract the second term from the first term:
\[ 5.25 - 3/4 \]
To simplify this, we need to convert 3/4 to a decimal. Dividing 3 by 4, we get 0.75.
\[ 5.25 - 0.75 = 4.50 \]
So, the common difference \(d\) between consecutive terms is 4.50.
Now that we have both \(a_1\) and \(d\), we can use the formula to find the 100th term:
\[ a_{100} = 3/4 + (100-1) \cdot 4.50 \]
Simplifying this equation gives us:
\[ a_{100} = 3/4 + 99 \cdot 4.50 \]
To find the 100th term, we need to convert \(3/4\) to a decimal. Dividing 3 by 4, we get 0.75.
\[ a_{100} = 0.75 + 99 \cdot 4.50 \]
Calculating this expression gives us:
\[ a_{100} = 0.75 + 445.50 = 446.25 \]
So, the 100th term of the arithmetic sequence is 446.25, rounding to the nearest hundredth.