Margot and Francois were doing their algebra homework when Margot's dog, Pepe, bit off a piece of the paper they were working on. All they had left was a part of the sequence: 6, 12, ... and the fact that 1536 was another term of the sequence. Assuming the sequence is arithmetic, what term number is 1536 ?
Second Part
Assuming the sequence is geometric, what term number is 1536 ?
arithmetic ... the difference is 6
(1536 - 12) / 6 = ?
12 is the 2nd term ... 1536 is the (?+2) term
geometric ... the ratio is 2
12 * 2^? = 1536
1536 is the (? + 2) term
thx
To find the term number of 1536 in the arithmetic sequence, we can 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.
In this case, we have the first term \(a_1 = 6\) and another term \(a_n = 1536\). We need to find \(n\), the term number. We also know that the sequence is arithmetic, which means there is a constant difference between consecutive terms.
To find the common difference (\(d\)), we can use the second term (\(a_2 = 12\)) and the first term (\(a_1 = 6\)):
\[d = a_2 - a_1\]
\[d = 12 - 6\]
\[d = 6\]
Now, we can substitute the values we know into the formula and solve for \(n\):
\[1536 = 6 + (n - 1) \times 6\]
Simplifying the equation:
\[1536 = 6 + 6n - 6\]
\[1536 = 6n\]
Now, divide both sides of the equation by 6:
\[\frac{1536}{6} = \frac{6n}{6}\]
\[256 = n\]
Therefore, the term number of 1536 in the arithmetic sequence is 256.