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
atroni [7]
3 years ago
5

Find the nth term of the sequence 7,25,51,85,127​

Mathematics
1 answer:
olya-2409 [2.1K]3 years ago
4 0

Let <em>a </em>(<em>n</em>) denote the <em>n</em>-th term of the given sequence.

Check the forward differences, and denote the <em>n</em>-th difference by <em>b </em>(<em>n</em>). That is,

<em>b </em>(<em>n</em>) = <em>a </em>(<em>n</em> + 1) - <em>a </em>(<em>n</em>)

These so-called first differences are

<em>b</em> (1) = <em>a</em> (2) - <em>a</em> (1) = 25 - 7 = 18

<em>b</em> (2) = <em>a</em> (3) - <em>a</em> (2) = 51 - 25 = 26

<em>b </em>(3) = <em>a</em> (4) - <em>a</em> (3) = 85 - 51 = 34

<em>b</em> (4) = <em>a </em>(5) - <em>a</em> (4) = 127 - 85 = 42

Now consider this sequence of differences,

18, 26, 34, 42, …

and notice that the difference between consecutive terms in this sequence <em>b</em> is 8:

26 - 18 = 8

34 - 26 = 8

42 - 34 = 8

and so on. This means <em>b</em> is an arithmetic sequence, and in particular follows the rule

<em>b</em> (<em>n</em>) = 18 + 8 (<em>n</em> - 1) = 8<em>n</em> + 10

for <em>n</em> ≥ 1.

So we have

<em>a </em>(<em>n</em> + 1) - <em>a </em>(<em>n</em>) = 8<em>n</em> + 10

or, replacing <em>n</em> + 1 with <em>n</em>,

<em>a</em> (<em>n</em>) = <em>a</em> (<em>n</em> - 1) + 8 (<em>n</em> - 1) + 10

<em>a</em> (<em>n</em>) = <em>a</em> (<em>n</em> - 1) + 8<em>n</em> + 2

We can solve for <em>a</em> (<em>n</em>) by iteratively substituting:

<em>a</em> (<em>n</em>) = [<em>a</em> (<em>n</em> - 2) + 8 (<em>n</em> - 1) + 2] + 8<em>n</em> + 2

<em>a</em> (<em>n</em>) = <em>a </em>(<em>n</em> - 2) + 8 (<em>n</em> + (<em>n</em> - 1)) + 2×2

<em>a</em> (<em>n</em>) = [<em>a</em> (<em>n</em> - 3) + 8 (<em>n</em> - 2) + 2] + 8 (<em>n</em> + (<em>n</em> - 1)) + 2×2

<em>a</em> (<em>n</em>) = <em>a</em> (<em>n</em> - 3) + 8 (<em>n</em> + (<em>n</em> - 1) + (<em>n</em> - 2)) + 3×2

and so on. The pattern should be clear; we end up with

<em>a</em> (<em>n</em>) = <em>a</em> (1) + 8 (<em>n</em> + (<em>n</em> - 1) + … + 3 + 2) + (<em>n</em> - 1)×2

The middle group is the sum,

\displaystyle 8\sum_{k=2}^nk=8\sum_{k=1}^nk-8=\frac{8n(n+1)}2-8=4n^2+4n-8

so that

<em>a</em> (<em>n</em>) = <em>a</em> (1) + (4<em>n</em> ² + 4<em>n</em> - 8) + 2 (<em>n</em> - 1)

<em>a</em> (<em>n</em>) = 4<em>n</em> ² + 6<em>n</em> - 3

You might be interested in
The equation |ax + b| = c must have ?
ArbitrLikvidat [17]

Answer:

0, 1, or 2 solutions

Step-by-step explanation:

The equation |ax + b| = c

The equation have absolute value symbol

Absolute value always gives us the positive number.

For absolute value function , we need to consider two cases

positive and negative.

|x|=x  for positive ,  and |-x|=x for negative case

For negative case we include negative sign

So  |ax + b| = c can be written as 2 equations

(ax+b)=c                                  (ax+b)=-c

So we will get maximum of 2 solutions

The equation |ax + b| = c must have 0, 1, or 2 solutions

4 0
3 years ago
Read 2 more answers
What is a name for a symbol like +, -, ∙ or ÷?
Radda [10]
+ is positive, or plus
- is negative, or minus
the third one is multiply, can look like * and "x"
the fourth one is divide, can look like "/"

hope this helps
8 0
3 years ago
Read 2 more answers
Find the interest accumulated from $100 dollars, with a 5% interest rate per month, for 2 months.
Sergio039 [100]
I believe you get 10 dollars in interest.
6 0
3 years ago
Read 2 more answers
Anthony has a sink that is shaped like a half-sphere. The sink has a volume of 4000/3*π in^3. One day, his sink clogged. He has
Nostrana [21]

That's a huge one. I don't think that anyone will actually answer that.

7 0
3 years ago
Find the equation of the line with Slope = -4 and passing through (-1,8). Write your equation in the form
Thepotemich [5.8K]

Answer:

y= -4x+4

Step-by-step explanation:

y=mx+b

where x=-1, y=8, and m=-4

8=(-4)(-1)+b

8=4+b

4=b

Thus, y= -4x+4

8 0
2 years ago
Other questions:
  • Yolanda is making a banner for a school pep rally. She cuts fabric in the shape of a parallelogram. The angle at the bottom left
    9·2 answers
  • -7.71875 as mixed number in simplest form
    15·1 answer
  • What is the answer to this equation?
    12·1 answer
  • Yall imma fail if you don't help me. what is the scientific notation for 0.12? I need answers asap
    6·2 answers
  • Santos flipped a coin 300 times. The coin landed heads up 175 times. Find the ratio of heads to total number of coin flips. Expr
    10·1 answer
  • Based on the graph below, what is a reasonable estimate for the у value when x is 4?
    7·1 answer
  • What does “B” equal in degrees?
    6·1 answer
  • What is the less of side a? Round to the nearest tenth of an inch.
    15·1 answer
  • Equations<br> 6) 2x + 3y = -1<br> 3x + 4y = -4
    11·1 answer
  • PLEASE HELP WILL GIVE YOU ALL MY POINTS In the picture, t || s, mz1 = 10x + 2, and mz2 = 12x - 22. Find the measure of angle 2.
    13·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!