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frutty [35]
1 year ago
14

Please help me find the answers to all the questions whoever answers all the question I will mark the brainlest.

Mathematics
1 answer:
Alex73 [517]1 year ago
7 0

Answer:

\textsf{a)} \quad \textsf{Recursive Rule}: \quad a_n=5n-2

     \textsf{Explicit Rule}: \quad \begin{cases}a_1=3\\a_n=a_{n-1}+5\end{cases}

b)  4998

c)  explicit rule (see below for explanation)

d)  see below

Step-by-step explanation:

An explicit formula for an <u>arithmetic sequence</u> allows you to find the nth term of the sequence.

A recursive formula for an <u>arithmetic sequence</u> allows you to find the nth term of the sequence <em>provided </em>you know the value of the previous term in the sequence.

<h3><u>Part a</u></h3>

<u>Explicit Formula</u>

a_n=a+(n-1)d

where:

  • a_n is the nth term
  • a is the first term
  • n is the number of the term
  • d is the common difference

<u>Given sequence</u>:  3, 8, 13, 18, 23, ...

To find the <u>common difference</u>, subtract one term from the next term.

Therefore:

  • a = 3
  • d = 8 - 3 = 5

Substituting the found values of a and d into the formula to create a explicit rule for the given sequence:

\implies a_n=3+(n-1)5

\implies a_n=3+5n-5

\implies a_n=5n-2

<u>Recursive Formula</u>

a_n=a_{n-1}+d \quad \textsf{for }n\geq 2

where:

  • a_n  is the nth term
  • a_{n-1} is the term immediately before the nth term
  • d is the common difference

We already know the common difference our previous calculations.

Therefore:

\implies a_n=a_{n-1}+5

When giving a recursive rule we also have to define the <u>first term</u> of the sequence, as it is not part of the formula.  Therefore, the full recursive rule for the given sequence is:

  \begin{cases}a_1=3\\a_n=a_{n-1}+5\end{cases}

<h3><u>Part b</u></h3>

To find the 1000th term of the sequence, use the <u>explicit rule</u> and substitute n = 1000:

\implies a_{1000}=5(1000)-2

\implies a_{1000}=5000-2

\implies a_{1000}=4998

Therefore, the 1000th term of the sequence is 4998.

<h3><u>Part c</u></h3>

We used the explicit rule to find the 1000th term as this rule allows us to find the nth term of the sequence <em>without </em>having to know any previous terms.

<h3><u>Part d</u></h3>

The <u>disadvantage</u> of a recursive rule when compared to an explicit rule is that we need to know the <u>previous term</u> to find the nth term, whereas the explicit rule doesn't need this information.

Learn more about sequences here:

brainly.com/question/27924553

brainly.com/question/27775450

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