Recognisable Pattern

Number Theory Level 2

Observe the pattern

2, 3, 6, 11,...

Which of the following is the formula for the nth term?

{ 2n }^{ 2 }-5n+5

((n-1)th term)((n-2)th term)

\frac { { 3n }^{ 2 }-7n+8 }{ 2 }

{ n }^{ 2 }-2n+3

5 solutions

Arunkumar Rajavel
Jul 28, 2014

The formula doesn't hold true for n=0 and for negative numbers. The post must include the term 'natural numbers' in it.

But a pattern cannot have terms like the -1st term or the 0th term.

K. J. W. - 6 years, 10 months ago

6^2=36, 36 - (2*6)=24, 24 + 3 = 27

Scott Hornberger - 6 years, 10 months ago

Sum=1+2+3+6+11+18....... -Sum= 1+2+3+ 6 + 11........nth term =: 0= 1+1+1+3+5+7.............-nth term nth term = 3+(3+5+7+....... n-2 terms) nth term = 3+(((n-2)/2)) 6+(n-3)2) nth term = n^2-2n+3

Himanshu Sharma - 6 years, 10 months ago

I was talking about n^2 - 2n + 3

Arunkumar Rajavel - 6 years, 10 months ago

According to traditional definition of natural number, 0 doesn't include into the series

jitendra kumar - 6 years, 10 months ago

meaning the way the problem is set up, in a very stick def, one has to specify for n that belongs to set of integers, along with 0.

Yuliya Skripchenko - 6 years, 10 months ago

As it is a sequence there is no need to specify that n is a natural number. We can have only 1 st, 2nd, 3rd , etc ., terms only.

Brahmam Meka - 6 years, 7 months ago

K. J. W.
Jul 28, 2014

Instinctively, some people would only bother checking the first two terms and of course, get the wrong answer as two of the answers are correct for only the first two terms.

Checking n^2-2n+3, we can see that all the numbers are correct.

2n^2-5n+5 is wrong: substitute n=3 and we get 8.

(3n^2-7n+8)/2 is wrong: substitute n=3 and we get 7.

((n-1)th term)((n-2)th term) is wrong: substitute n=4 and we get 18.

Technically, yes, there is a way.

Since ((n-1)th term)((n-2)th term) is wrong, the formula is a quadratic.

Then A+B+C=2

3A+B=1

2A=2

A=1

B=-2

C=3

K. J. W. - 6 years, 10 months ago

Well, you need to go based off the pattern. 4 is not n because 4 is not a given number in the pattern. Many numbers will be skipped, but they are none of our concern. I believe the correct answer is wrong though. While n=2 gives us 3 and n=3 gives us 6, n=6 does not equal 11. (6)^2-2(6)+3= 36-12+3= 24+3= 27 27 does not equal 11 (obviously)

Randie Girard - 6 years, 10 months ago

The "nth term" is not the previous term in the sequence, it is the number of the term.

Shawn Ong - 6 years, 10 months ago

I think that you misunderstood the question. 11 is the 4th term and not the 6th term .

K. J. W. - 6 years, 10 months ago

Clarification once and for all:

2 is the 1st term

3 is the 2nd term

6 is the 3rd term

11 is the 4th term

K. J. W. - 6 years, 10 months ago

You don't just use checking as a solution. There must be a way to get the nth term.

Mark Angelo Ordonio - 6 years, 10 months ago

It is easy to solve. The way you prove it is this: First you have to figure out if it is a polynomial. To do that we look at the patterns between the numbers. The difference between 2-3 is 1, 3-6 is 3, 6-11 is 5. Now we have the set 1, 3, 5. We can do this again and between 1-3 is 2, and 3-5 is also 2. Now that we have a constant number, it means that if we were to derive this equation twice we would get a constant meaning that it is polynomial and it is also quadratic. Once we know it is quadratic it is pretty simple on the rest. So because the constant slope is 2, we integrate two twice and get X^2 + Cx + K. Now luckily for us there is only one x^2 solution so we know it is correct but we can keep proving. So the constant k is the value of the sequence when x = 0 (side note I am using x instead of n). We can very easily figure out when x is 0 by looking at the slope pattern. So we know that 2 is our slope for our second derivative, this means that the slope between set 0 and 1 is going to be -1 (1-2=-1). So now we simply subtract our change in value (-1) from the first term 2, and we will get our k value. The k is equal to 3 as 2 - (-1)=3. so know we have the equation x^2 +Cx +3. Now the tricky parts are over and it is a matter of plugging in one of your values to figure out constant c. So lets do term 1. (1)^2 +c(1) + 3 = 2. 4+c = 2, c=-2. Now we have the general equation X^2 - 2x + 3

Henry Delargy - 6 years, 10 months ago

You must check your calculations again as the answer after substituting 6 is 27 and not 11. I tired all the forms but non of them was correct and I think you should revise your answer again.

A Former Brilliant Member - 6 years, 10 months ago

Sumit Saha
Sep 4, 2014

s=2+3+6+11+.........

s= 2+3+6+11+.......(by shifting)

0=2+1+3+5+........nth term-tn tn=2+(1+3+5+....n--1 term) tn=2+(n-1)(n-1) tn=n.n-2.n+3

Nikhil Tandon
Aug 17, 2014

Put n=4. Only the 3rd option will give 11 as answer.

Dladla Arthur
Aug 7, 2014

2,3,6,11

1st diferrence : 1, 3, 5

2nd differrence:

2a = 2 therefore a =1

3a + b = 1 therefore b = -2

a+b+c = 2 therefore c = 3

By using the formula Tn = an^2 + bn + c

we obtain Tn = n^2 - 2n + 3

because 2nd difference is constant it is a quadratic, other wise great explanation

Yuliya Skripchenko - 6 years, 10 months ago

Recognisable Pattern

5 solutions

s= 2+3+6+11+.......(by shifting)

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