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
coldgirl [10]
2 years ago
8

Determine formula of the nth term 2, 6, 12 20 30,42​

Mathematics
1 answer:
nalin [4]2 years ago
6 0

Check the forward differences of the sequence.

If \{a_n\} = \{2,6,12,20,30,42,\ldots\}, then let \{b_n\} be the sequence of first-order differences of \{a_n\}. That is, for n ≥ 1,

b_n = a_{n+1} - a_n

so that \{b_n\} = \{4, 6, 8, 10, 12, \ldots\}.

Let \{c_n\} be the sequence of differences of \{b_n\},

c_n = b_{n+1} - b_n

and we see that this is a constant sequence, \{c_n\} = \{2, 2, 2, 2, \ldots\}. In other words, \{b_n\} is an arithmetic sequence with common difference between terms of 2. That is,

2 = b_{n+1} - b_n \implies b_{n+1} = b_n + 2

and we can solve for b_n in terms of b_1=4:

b_{n+1} = b_n + 2

b_{n+1} = (b_{n-1}+2) + 2 = b_{n-1} + 2\times2

b_{n+1} = (b_{n-2}+2) + 2\times2 = b_{n-2} + 3\times2

and so on down to

b_{n+1} = b_1 + 2n \implies b_{n+1} = 2n + 4 \implies b_n = 2(n-1)+4 = 2(n + 1)

We solve for a_n in the same way.

2(n+1) = a_{n+1} - a_n \implies a_{n+1} = a_n + 2(n + 1)

Then

a_{n+1} = (a_{n-1} + 2n) + 2(n+1) \\ ~~~~~~~= a_{n-1} + 2 ((n+1) + n)

a_{n+1} = (a_{n-2} + 2(n-1)) + 2((n+1)+n) \\ ~~~~~~~ = a_{n-2} + 2 ((n+1) + n + (n-1))

a_{n+1} = (a_{n-3} + 2(n-2)) + 2((n+1)+n+(n-1)) \\ ~~~~~~~= a_{n-3} + 2 ((n+1) + n + (n-1) + (n-2))

and so on down to

a_{n+1} = a_1 + 2 \displaystyle \sum_{k=2}^{n+1} k = 2 + 2 \times \frac{n(n+3)}2

\implies a_{n+1} = n^2 + 3n + 2 \implies \boxed{a_n = n^2 + n}

You might be interested in
Which of these professionals most directly uses geometry?
notka56 [123]
Good Morning!
<span>air traffic controller
</span>

The most cited professional who uses geometry is the air traffic controller. This professional needs to calculate routes, distances and times traveled in order to avoid accumulation of aircraft at the time of landing, for example.
8 0
2 years ago
Find P(x)=... polynomial
Aneli [31]

Answer: D) P(x) = -23

Step-by-step explanation:

Substitute 2 for x:

1 - 2(2)^2-3(2)^2+4(2)

Exponent

1-2(4)-3(8)+4(2)

Multiply

1-8-24+8

Combine like terms

-23

<em>Hope it helps <3</em>

3 0
2 years ago
In the diagram below, line AC intersects triangle BDE at B. Choose True or False for each statement.
mixer [17]

Answer:

t'

f

t

Step-by-step explanation:

6 0
2 years ago
Today, both the soccer team and the basket ball team had games (starting on Monday). The soccer team plays every 3 days and the
yarga [219]
I think eight because if you add those days together they will play again in eight days so that they play on the same day.

Sorry if I got it wrong I tried and that's what counts, right?
3 0
2 years ago
Pleas help me with this answer https://brainly.com/question/19857050
mylen [45]
What ??????? !.9:9..jdkKzksoq
3 0
2 years ago
Other questions:
  • Select True or False for each statement. (10pts)
    13·2 answers
  • X over 15 = 10 over 20
    5·1 answer
  • The local bike shop sells a bike and accessories package for $320 if the bike is worth 7 times more than the accessories,how muc
    8·1 answer
  • The perimeter of a rectangle pool is 160 yards. if the length is 10 yards less than twice the width, find the length and the wid
    6·1 answer
  • Help out dudes!! #38.
    5·2 answers
  • 1.5÷5.46 show your work
    12·1 answer
  • Kyle has $1500 to invest in a saving's account for 5 years. He has 3 options:
    12·1 answer
  • Which expression are equal to the value of 3 in the number 14,397
    12·1 answer
  • Sara had 27 peaches and 11 pears left at her roadside fruit stand. She went to the orchard and picked more peaches to stock up t
    6·2 answers
  • PLEASE HELP ME
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!