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On October 1, Nadia starts a push-up challenge by doing 18 push-ups. On October 2, she does 21 push¬ups. On October 3, she does 24 push-ups. She continues until October 16, when she does the final push-ups in the challenge. a. Write an explicit definition to model the number of push-ups Nadia does each day. b. Write a recursive definition to model the number of push-ups Nadia does each day. c. How many push-ups will Nadia do on October 16? d. What is the total number of push-ups Nadia does from October 1 to October 16?



Answer :

semsee45

Answer:

[tex]\textsf{a)} \quad a_n=3n+15[/tex]

[tex]\textsf{b)} \quad \begin{cases}a_n=a_{n-1}+d\\a_1=18\end{cases}[/tex]

[tex]\textsf{c)} \quad 63[/tex]

[tex]\textsf{d)} \quad 648[/tex]

Step-by-step explanation:

Part (a)

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

Explicit Formula

[tex]\boxed{a_n=a+(n-1)d}[/tex]

where:

  • [tex]a_n[/tex] is the nth term.
  • a is the first term.
  • n is the number of the term.
  • d is the common difference.

Given information:

  • October 1 = 18 push-ups
  • October 2 = 21 push-ups
  • October 3 = 24 push-ups

Nadia increases the number of push-ups each day by 3.  Therefore:

  • a = 18
  • d = 3

Substitute the values of a and d into the formula to create an explicit formula to model the number of push-ups Nadia does each day:

[tex]\implies a_n=18+(n-1)3[/tex]

[tex]\implies a_n=18+3n-3[/tex]

[tex]\implies a_n=3n+15[/tex]

Part (b)

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

Recursive Formula

[tex]\boxed{a_n=a_{n-1}+d}[/tex]

where:

  • [tex]a_n[/tex] is the nth term.
  • [tex]a_{n-1}[/tex] is the (n-1)th term.
  • d is the common difference.

We already know the common difference from the previous calculations.

Therefore:

[tex]\implies a_n=a_{n-1}+3[/tex]

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

[tex]\begin{cases}a_n=a_{n-1}+d\\a_1=18\end{cases}[/tex]

Part (c)

To calculate how many push-ups Nadia will do on October 16, substitute n = 16 into the recursive formula from part (a):

[tex]\implies a_{16}=3(16)+15[/tex]

[tex]\implies a_{16}=48+15[/tex]

[tex]\implies a_{16}=63[/tex]

Therefore, Nadia will do 63 push-ups on October 16.

Part (d)

Sum of the first n terms of an arithmetic series:

[tex]\boxed{S_n=\dfrac12n(a+a_n)}[/tex]

To find the total number of push-ups Nadia does from October 1 to October 16, substitute n = 16, a = 18 and a₁₆ = 63 into the formula:

[tex]\implies S_{16}=\dfrac12(16)(18+63)[/tex]

[tex]\implies S_{16}=8(81)[/tex]

[tex]\implies S_{16}=648[/tex]

Therefore, Nadia does a total number of 648 push-ups from October 1 to October 16.

Learn more about arithmetic sequences here:

https://brainly.com/question/28010120

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