Math

QuestionFind the first 5 terms of the sequence an=5nn+6a_{n}=\frac{5 n}{n+6} and compute limnan\lim _{n \rightarrow \infty} a_{n}.

Studdy Solution

STEP 1

Assumptions1. The sequence is defined by the formula an=5nn+6a_{n}=\frac{5n}{n+6} for n=0,1,,3,n=0,1,,3,\ldots . We are asked to find the first5 terms of the sequence and the limit of the sequence as nn approaches infinity.

STEP 2

To find the first5 terms of the sequence, we substitute the values n=0,1,2,,4n=0,1,2,,4 into the formula.

STEP 3

Substitute n=0n=0 into the formula to find the first term.
a0=5(0)0+6a_{0}=\frac{5(0)}{0+6}

STEP 4

Calculate the first term.
a0=06=0a_{0}=\frac{0}{6}=0

STEP 5

Substitute n=1n=1 into the formula to find the second term.
a1=5(1)1+a_{1}=\frac{5(1)}{1+}

STEP 6

Calculate the second term.
a1=5a_{1}=\frac{5}{}

STEP 7

Substitute n=2n=2 into the formula to find the third term.
a2=5(2)2+6a_{2}=\frac{5(2)}{2+6}

STEP 8

Calculate the third term.
a2=108=54a_{2}=\frac{10}{8}=\frac{5}{4}

STEP 9

Substitute n=3n=3 into the formula to find the fourth term.
a3=5(3)3+6a_{3}=\frac{5(3)}{3+6}

STEP 10

Calculate the fourth term.
a3=159=53a_{3}=\frac{15}{9}=\frac{5}{3}

STEP 11

Substitute n=4n=4 into the formula to find the fifth term.
a4=5(4)4+6a_{4}=\frac{5(4)}{4+6}

STEP 12

Calculate the fifth term.
a4=2010=2a_{4}=\frac{20}{10}=2

STEP 13

The first5 terms of the sequence are 0,57,5,53,20, \frac{5}{7}, \frac{5}{}, \frac{5}{3},2.

STEP 14

To find the limit of the sequence as nn approaches infinity, we take the limit of the formula as nn approaches infinity.
limnan=limnnn+6\lim{n \rightarrow \infty} a_{n} = \lim{n \rightarrow \infty} \frac{n}{n+6}

STEP 15

We can simplify this limit by dividing the numerator and denominator by nn.
limnan=limn5+n\lim{n \rightarrow \infty} a_{n} = \lim{n \rightarrow \infty} \frac{5}{+\frac{}{n}}

STEP 16

As nn approaches infinity, 6n\frac{6}{n} approaches0.
limnan=5+0\lim{n \rightarrow \infty} a_{n} = \frac{5}{+0}

STEP 17

Calculate the limit.
limnan=5\lim{n \rightarrow \infty} a_{n} =5The first5 terms of the sequence are 0,57,54,53,20, \frac{5}{7}, \frac{5}{4}, \frac{5}{3},2 and the limit of the sequence as nn approaches infinity is5.

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