20. Sequences, Series & Induction

Sequences

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Multiple Choice

Determine the first 3 terms of the sequence given by the general formula
$a_{n}=\frac{1}{n!+1}$

$\left\lbrace\frac12,\frac13,\frac17\right\rbrace$

$\left\lbrace\frac12,\frac13,\frac14\right\rbrace$

$\left\lbrace1,2,7\right\rbrace$

$\left\lbrace1,\frac12,\frac16\right\rbrace$

Verified step by step guidance

Identify the general formula for the sequence: a_n = \frac{1}{n! + 1}.

To find the first term (a_1), substitute n = 1 into the formula: a_1 = \frac{1}{1! + 1}.

Calculate 1! (factorial of 1), which is 1, and then compute 1! + 1.

Substitute the result into the formula to find a_1: a_1 = \frac{1}{2}.

Repeat the process for n = 2 and n = 3 to find a_2 and a_3, substituting n = 2 and n = 3 into the formula and calculating the factorials accordingly.

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Determine the first 3 terms of the sequence given by the general formulaan=1n!+1a_{n}=\frac{1}{n!+1}an=n!+11