Consider the arithmetic sequence 13, 24, 35, ....

Mathematics

1 answer:

MArishka [77]3 years ago

8 0

Answer:

The explicit form for the sequence is: $a_{n}=13+(n-1)(11)$

Step-by-step explanation:

In order to find an explicit form for the given sequence, you have to use the definition of arithmetic sequence and the explicit formula.

An arithmetic sequence is defined as a sequence where the difference of two consecutive terms is a constant.

The explicit formula is:

$a_{n}=a_{1} + (n-1)d$

Where a1 is the first term, d is the common difference and an is the nth term of the sequence.

You have to subtract two consecutive terms to obtain d:

24-13= 11

35-24=11

Therefore d=11

In this case a1=13

Replacing in the formula:

$a_{n}=13+(n-1)(11)$

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James and Brandon both bought a drink from the store. James drank 3/4 of his drink. How many eights did brandon drink?

maksim [4K]

Answer:

6/8

Step-by-step explanation:

3/4.

3×2=6.

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3 years ago

Write the polynomial in factored form as a product of linear factors f(r)=r^3-9r^2+17r-9

adelina 88 [10]

Answer:

f(r) = (x -1)(x -4+√7)(x -4-√7)

Step-by-step explanation:

The signs of the terms are + - + -. There are 3 changes in sign, so Descartes' rule of signs tells you there are 3 or 1 positive real roots.

The rational roots, if any, will be factors of 9, the constant term. The sum of coefficients is 1 -9 +17 -9 = 0, so you know that r=1 is one solution to f(r) = 0. That means (r -1) is a factor of the function.

Using polynomial long division, synthetic division (2nd attachment), or other means, you can find the remaining quadratic factor to be r^2 -8r +9. The roots of this can be found by various means, including completing the square:

r^2 -8r +9 = (r^2 -8r +16) +9 -16 = (r -4)^2 -7

This is zero when ...

(r -4)^2 = 7

r -4 = ±√7

r = 4±√7

Now, we know the zeros are {1, 4+√7, 4-√7), so we can write the linear factorization as ...

f(r) = (r -1)(r -4 -√7)(r -4 +√7)

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Comment on the graph

I like to find the roots of higher-degree polynomials using a graphing calculator. The red curve is the cubic. Its only rational root is r=1. By dividing the function by the known factor, we have a quadratic. The graphing calculator shows its vertex, so we know immediately what the vertex form of the quadratic factor is. The linear factors are easily found from that, as we show above. (This is the "other means" we used to find the quadratic roots.)