Math  /  Algebra

QuestionGive a formula for the sequence given by 21,42,86,1624,\frac{2}{1}, \frac{4}{2}, \frac{8}{6}, \frac{16}{24}, \ldots

Studdy Solution

STEP 1

1. The sequence is given as a series of fractions.
2. The numerators and denominators follow a specific pattern.
3. We need to find a general formula for the n n -th term of the sequence.

STEP 2

1. Identify the pattern in the numerators.
2. Identify the pattern in the denominators.
3. Combine the patterns to form a general formula for the sequence.

STEP 3

Observe the numerators: 2,4,8,16, 2, 4, 8, 16, \ldots .
The numerators follow a pattern of powers of 2: 21,22,23,24, 2^1, 2^2, 2^3, 2^4, \ldots .
Thus, the general formula for the numerator of the n n -th term is 2n 2^n .

STEP 4

Observe the denominators: 1,2,6,24, 1, 2, 6, 24, \ldots .
The denominators follow a pattern of factorials: 1!,2!,3!,4!, 1!, 2!, 3!, 4!, \ldots .
Thus, the general formula for the denominator of the n n -th term is n! n! .

STEP 5

Combine the formulas for the numerators and denominators.
The n n -th term of the sequence is given by:
an=2nn! a_n = \frac{2^n}{n!}
The formula for the sequence is:
2nn! \boxed{\frac{2^n}{n!}}

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