1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Marianna [84]
1 year ago
11

What is the nth term rule of the quadratic sequence below?

Mathematics
1 answer:
Vladimir [108]1 year ago
6 0

Answer:

3n² + 5n - 2

Step-by-step explanation:

<u>Given sequence</u>:

6, 20, 40, 66, 98, 136, ...

Calculate the <u>first differences</u> between the terms:

6 \underset{+14}{\longrightarrow} 20 \underset{+20}{\longrightarrow} 40 \underset{+26}{\longrightarrow} 66 \underset{+32}{\longrightarrow} 98 \underset{+38}{\longrightarrow} 136

As the first differences are not the same, calculate the <u>second differences:</u>

14 \underset{+6}{\longrightarrow} 20 \underset{+6}{\longrightarrow} 26 \underset{+6}{\longrightarrow} 32 \underset{+6}{\longrightarrow} 38

As the <u>second differences are the same</u>, the sequence is quadratic and will contain an n² term.

The <u>coefficient</u> of the n² term is <u>half of the second difference</u>.

Therefore, the n² term is:  3n²

Compare 3n² with the given sequence:

\begin{array}{|c|c|c|c|c|}\cline{1-5} n & 1 & 2 & 3 & 4\\\cline{1-5} 3n^2 & 3 & 12 & 27 & 48 \\\cline{1-5} \sf operation & +3&+8 & +13 & +18 \\\cline{1-5} \sf sequence & 6 & 20 & 40 & 66\\\cline{1-5}\end{array}

The second operations are different, therefore calculate the differences <em>between</em> the second operations:

3 \underset{+5}{\longrightarrow} 8 \underset{+5}{\longrightarrow} 13\underset{+5}{\longrightarrow} 18

As the differences are the same, we need to add 5n as the second operation:

\begin{array}{|c|c|c|c|c|}\cline{1-5} n & 1 & 2 & 3 & 4\\\cline{1-5} 3n^2  +5n & 8&22 & 42 & 68\\\cline{1-5}\sf operation & -2 &-2  &-2  & -2  \\\cline{1-5} \sf sequence & 6 & 20 & 40 & 66\\\cline{1-5}\end{array}

Finally, we can clearly see that the operation to get from 3n² + 5n to the given sequence is to subtract 2.

Therefore, the nth term of the quadratic sequence is:

3n² + 5n - 2

You might be interested in
How do you write out.32ft/sec to meters/min
marta [7]
32 ft 
this is meters per second >>> 9.7536
6 0
3 years ago
- 3 + 3x = 2x – 13 <br><br> I have to use more characters...
DENIUS [597]

- 3 + 3x = 2x – 13

Bring -3 to the right side of the equation by adding 3 to both sides

(- 3 + 3) + 3x = 2x – 13 + 3

0 + 3x = 2x - 10

3x = 2x - 10

Bring 2x to the other side by subtracting 2x to both sides

3x - 2x = 2x - 2x - 10

x = -10

Check:

-3 + 3(-10) = 2(-10) - 13

-3 + (-30) = -20 - 13

-33 = -33

Hope this helped!

~Just a girl in love with Shawn Mendes

6 0
2 years ago
<img src="https://tex.z-dn.net/?f=%20%5Crm%20%5Cint_%7B0%7D%5E%7B%20%20%5Cpi%20%7D%20%5Ccos%28%20%5Ccot%28x%29%20%20%20%20-%20%2
Nikolay [14]

Replace x with π/2 - x to get the equivalent integral

\displaystyle \int_{-\frac\pi2}^{\frac\pi2} \cos(\cot(x) - \tan(x)) \, dx

but the integrand is even, so this is really just

\displaystyle 2 \int_0^{\frac\pi2} \cos(\cot(x) - \tan(x)) \, dx

Substitute x = 1/2 arccot(u/2), which transforms the integral to

\displaystyle 2 \int_{-\infty}^\infty \frac{\cos(u)}{u^2+4} \, du

There are lots of ways to compute this. What I did was to consider the complex contour integral

\displaystyle \int_\gamma \frac{e^{iz}}{z^2+4} \, dz

where γ is a semicircle in the complex plane with its diameter joining (-R, 0) and (R, 0) on the real axis. A bound for the integral over the arc of the circle is estimated to be

\displaystyle \left|\int_{z=Re^{i0}}^{z=Re^{i\pi}} f(z) \, dz\right| \le \frac{\pi R}{|R^2-4|}

which vanishes as R goes to ∞. Then by the residue theorem, we have in the limit

\displaystyle \int_{-\infty}^\infty \frac{\cos(x)}{x^2+4} \, dx = 2\pi i {} \mathrm{Res}\left(\frac{e^{iz}}{z^2+4},z=2i\right) = \frac\pi{2e^2}

and it follows that

\displaystyle \int_0^\pi \cos(\cot(x)-\tan(x)) \, dx = \boxed{\frac\pi{e^2}}

7 0
2 years ago
A rectangular box has dimensions 12 by 16 by 21, as shown here:
professor190 [17]

Answer: 29

Step-by-step explanation:

d = √(l2 + w2 + h2)

d = √(12^2+ 16^2+ 21^2) = 29

3 0
2 years ago
I'll mark brainliest thanks in advance ​
beks73 [17]

Answer:

i think it's either 31 or 28

5 0
3 years ago
Read 2 more answers
Other questions:
  • Please answer and show a little work
    6·1 answer
  • How to do this question plz ​
    6·2 answers
  • Estimate the sum. 10 1/9 + 5 14/15
    15·1 answer
  • A hovercraft takes off from a platform. Its height (in meters), 2 seconds after takeoff, is modeled by: h(x) = -3(x - 3)2 + 108
    8·2 answers
  • 8 • 10 exponent 9 ÷ 1.2 • 10 exponent 6
    13·1 answer
  • Students were surveyed about their favorite colors. 1/4 of the students preffered red, 1/8 of the students preffed blue, and 3/5
    7·1 answer
  • Can somebody help me with this problem
    13·2 answers
  • Determine which function(s) is not a linear function. Select all that apply.
    6·1 answer
  • (2x2 – 18)(x+3)(2 – 3) = 2x4 – 36x2 + 162
    13·2 answers
  • When forests are replaced with cities,
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!