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4 years ago

Arithmetic sequences (nth term)

Mathematics

1 answer:

vodomira [7]4 years ago

6 0

Tick the arithmetic sequences.

In order for a sequence to be arithmetic the relations between their numbers must be the sum of a common ratio, therefore to determine which ones are arithmetic sequence we need to satisfy the following expression:

$a_n - a_{n-1} = a_{n-2} - a_{n-3}$

The expression just mean that the subtraction of consecutive terms should be equal for position in the sequence.

"1, 5, 9, 13...":

$13 - 9 = 5 - 1\\4 = 4$

Since the expression is valid, then this is a arithmetic sequence.

"6, 10, 15, 21...:

$21 - 15 = 10 - 6\\6 = 4$

Since the expression is invalid, then this isn't a arithmetic sequence.

"2, 3, 5, 8...:

$8 - 5 = 3 - 2\\3 = 1$

Since the expression is invalid, then this isn't a arithmetic sequence.

"2, -4, 8, -16...:

$-16- 8 = -4 - 2\\-24 = -6$

Since the expression is invalid, then this isn't a arithmetic sequence.

"-1, 2, 5, 8...:

$8 - 5 = 2 -(- 1)\\3 = 3$

Since the expression is valid, then this is a arithmetic sequence.

"73, 66, 59, 52...:

$52 - 59 = 66 - 73\\-7 = -7$

Since the expression is valid, then this is a arithmetic sequence.

"6, 1, -4, -9...:

$-9 - (-4) = 1 - 6\\-5 = -5$

Since the expression is valid, then this is a arithmetic sequence.

"1, 2, 4, 8...:

$8 - 4 = 2 - 1\\4 = 1$

Since the expression is invalid, then this isn't a arithmetic sequence.

Find the first five terms of the patterns with these nth terms.

For $3n + 3$ :

$3*1 + 3 = 3 + 3 = 6$

$6 + 3 = 9$

$9 + 3 = 12$

$12 + 3 = 15$

$15 + 3 = 18$

(6, 9, 12, 15, 18)

For $9n - 7$ :

$9*1 - 7 = 9 - 7 = 2$

$2 + 9 = 11$

$11 + 9 = 20$

$20 + 9 = 29$

$29 + 9 = 38$

(2,11,20,29,38)

Write down a formula for the nth term of these patterns. The first term is n = 1.

The nth term of any arithmetic sequence is: $a(n) = a(1)*(n - 1)*r$

Therefore we need to identify r in each sequence.

For (9,15,21,27,33):

$r = 15 - 9 = 6$

$a(n) = 1*(n - 1)*6 \\a(n) = 6*n - 6$

For (-2,-8,-14,-20,-26):

$r = -8 - (-2) = -6$

$a(n) = 1*(n - 1)*(-6)\\a(n) = -6*n + 6$

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A firework is fired into the air from the top of a barn. Tis hight (h) above the ground in yards after (t) seconds is given by t

Grace [21]

The given function for the height of the firework is a quadratic function

1. Time at which the firework reaches the maximum height is 1 seconds

2. The maximum height of the firework, is 25 yards

3. Time after which the firework will fall to the ground is (1 + √5) seconds

Reason:

The given function that represents the height of the fireworks with time is

presented as follows;

h(t) = -5·t² + 10·t + 20

1. The time at which the firework reaches its maximum height is given by

the maximum point of the given function as follows;

The x-value of the maximum point of a quadratic function is $x = \dfrac{b}{2 \cdot a}$

Where;

a, and b, are the coefficient of x² and x, in the general form of a quadratic function f(x) = a·x² + b·x + c

By comparison, we have;

$t = -\dfrac{10}{2 \times (-5)} = 1$

The time at which the firework reaches the maximum height is t = 1 seconds

2. The maximum height is given by plugging in the value of t, at the maximum point into the given function as follows;

h(1) = -5×1² + 10×1 + 20 = 25

The maximum height of the firework, f(1) = 25 yards

3 The time at which the firework will fall to the ground, is given by the zero of the function as follows;

When the firework falls to the ground, h(t) = 0 = -5·t² + 10·t + 20

Dividing both sides by (-5) gives;

$\dfrac{0}{-5} = \dfrac{ -5 \cdot t^2 + 10 \cdot t + 20}{-5} = t^2 - 2 \cdot t - 4$

t² - 2·t - 4 = 0

By the quadratic formula $x = \dfrac{-b\pm \sqrt{b^{2}-4\cdot a\cdot c}}{2\cdot a}$ , we get;

$t = \dfrac{2\pm \sqrt{(-2)^{2}-4\times 1\times (-4)}}{2\times 1} = \dfrac{2\pm \sqrt{20}}{2\times 1} = \dfrac{2\pm 2 \times \sqrt{5}}{2\times 1} = 1 \pm \sqrt{5}$