Respuesta :
Answer:
Write the first five terms of the sequence defined by the recursive formula.
a1=9an=3an−1−20, for n≥2" role="presentation" style="font-family: proxima-nova, sans-serif; -webkit-font-smoothing: subpixel-antialiased; padding: 1px 0px; margin: 0px; font-size: 15.696px; vertical-align: baseline; background: transparent; border: 0px; outline: 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">a1=9an=3an−1−20, for n≥2
The first term is given in the formula. For each subsequent term, we replace \displaystyle {a}_{n - 1}an−1 with the value of the preceding term.
n=1a1=9n=2a2=3a1−20=3(9)−20=27−20=7n=3a3=3a2−20=3(7)−20=21−20=1n=4a4=3a3−20=3(1)−20=3−20=−17n=5a5=3a4−20=3(−17)−20=−51−20=−71" role="presentation" style="font-family: proxima-nova, sans-serif; -webkit-font-smoothing: subpixel-antialiased; padding: 1px 0px; margin: 0px; font-size: 15.696px; vertical-align: baseline; background: transparent; border: 0px; outline: 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">n=1a1=9n=2a2=3a1−20=3(9)−20=27−20=7n=3a3=3a2−20=3(7)−20=21−20=1n=4a4=3a3−20=3(1)−20=3−20=−17n=5a5=3a4−20=3(−17)−20=−51
The recursive formula gives values based on the preceding term, while
the explicit formula is a general formula.
Correct response:
- Recursive formula; aₙ = a₍ₙ₋₁₎ - 5
- Explicit formula; aₙ = 35 - 5·(n - 1)
- a₅ = 15, a₆ = 10
How to write recursive and explicit formula
An example of the first four terms of a decreasing Arithmetic Sequence is presented as follows;
- 35, 30, 25, 20
Recursive formula:
From the above Arithmetic Sequence, we have;
The difference between each successive term is constant and equal to -5, therefore;
- The recursive formula is; [tex]\underline{a_n = a_{n -1} - 5}[/tex]
Explicit formula
The sequence of numbers can be taken as the output of a linear function where the input is the term of the number.
To find the equation of the linear function, we have;
[tex]The \ slope, \ m = \dfrac{30 - 35}{2 - 1} = -5 = d = The \ common \ difference[/tex]
The equation of the function in point and slope form is therefore;
aₙ - 35 = -5·(n - 1)
Which gives;
aₙ = 35 - 5·(n - 1)
Where;
aₙ = The value of the nth term of the Arithmetic Sequence
n = The term of the Arithmetic Sequence
35 = The first term of the sequence = a₁
-5 = The common difference = d
The explicit formula for the (a) Arithmetic Sequence is; aₙ = a₁ + (n - 1)·d
Which gives;
aₙ = 35 + (n - 1) × (-5) = 35 + (n - 1)·d
- The explicit formula is; [tex]\underline{a_n = 35 - 5 \times (n - 1)}[/tex]
The next two terms are found by using the recursive formulas as follows;
a₅ = a₄ - 5
a₆ = a₅ - 5
Which gives;
a₅ = 20 - 5 = 15
- Fifth term, a₅ = 15
a₆ = 15 - 5 = 10
- Sixth term, a₆ = 10
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