# Which number comes next in this series: 2, 5, 14, 41, 122, ?

Relevance
• Since you are only asking for the next number, an easy way looks like this:

The difference between each consecutive number is:

3, 9, 27, 81...

which are simply consecutive powers of 3

81 = 3^4

Then, the next interval will call for a difference of 3^5 = 81*3 = 243

Adding 243 to 122, we get 243+122 = 365

The next number after that is:

365 + 3^6 = 365 + 729 = 1094

2, 5, 14, 41, 122, 365, 1094, 5468, 18590, ...

This method was already well used in the 17th century (the 1600s) and may have started many centuries before that.

If you must find a much longer list of numbers (for example, if you are looking for the 99th number) then you must use more advanced formulas.

• The sequence is 3x - 1 where x is the previous number

To start: with x = 2

3x - 1

3(2) - 1

6 - 1

5

For 5

15 -1

14

For 14

3x - 1

3(14) - 1

42 - 1

41

For 41

123 - 1

122

FOR 122 ← QUESTION

3x - 1

3(122) - 1

366 - 1

The next number in the series is 365

• a(n) = 1/2 (3ⁿ + 1) (this was not easy to determine!)

yields 2, 5, 14, 41, 122, → 365 ← , 1094, 3281, 9842, 29525, 88574, 265721, 797162, etc.

So 365

• 2, 5, 14, 41, 122, 365.

• The nth term of that sequence could be:

n^4 - 8n^3 + 26n^2 - 34n + 17

In that case:

Term 6 is 317

• (x*3 - 1) , this series: 2, 5, 14, 41, 122,

• 2, 5, 14, 41, 122, ...

a_n = 1/2 (3^n + 1)

365 is the next number

• Each term is (3 * previous) - 1

Next term is (3 * 122) - 1 =  365

• Each increase is either three or a multiple of three times the difference between the previous and the higher number. For example, the difference between 5 and 2 is three. The difference between 14 and 5 is 9 or the previous difference multiplied by 3 and so on. This is a verifiable pattern. The difference between 122 and 41 is 81. 81 times 3 is 243. 243 plus 122 is 365. I have explained this rather than simply giving you the answer and that way you should understand the reasoning behind identifying patterns rather than just having one answer which is pretty useless for the most part. Give a person a fish and they eat for one day. Teach a person to fish and they can fee themselves for the rest of their life.

• t = 2

t = 3 * 2 - 1 = 3 * t - 1

t = 3 * 5 - 1 = 3 * t - 1

t = 3 * 14 - 1 = 3 * t - 1

t = 3 * 41 - 1 = 3 * t - 1

See a pattern?

t = 3 * t - 1 = 3 * (3 * t[1[ - 1) - 1 = 9 * t - 3 - 1 = 9 * t - 4

t = 3 * t - 1 = 3 * (9 * t - 4) - 1 = 27 * t - 12 - 1 = 27 * t - 13

t[n] = 3^(n - 1) * t - (1 + 3 + 3^2 + ... + 3^(n - 2))

S = 1 + 3 + 3^2 + ... + 3^(n - 2)

3S = 3 + 3^2 + 3^3 + ... + 3^(n - 1)

3S - S = 3^(n - 1) - 1

2S = (3^(n - 1) - 1)

S = (1/2) * (3^(n - 1) - 1)

t[n] = 3^(n - 1) * 2 - (1/2) * (3^(n - 1) - 1)

t[n] = 2 * 3^(n - 1) - (1/2) * 3^(n - 1) + (1/2)

t[n] = (3/2) * 3^(n - 1) + (1/2)

t[n] = (1/2) * (3 * 3^(n - 1) + 1)

t[n] = (1/2) * (3^(n) + 1)

t[n] = (3^(n) + 1) / 2

t = (3^1 + 1) / 2 = (3 + 1) / 2 = 4/2 = 2

t = (3^2 + 1) / 2 = (9 + 1) / 2 = 10/2 = 5

t = (3^3 + 1) / 2 = (27 + 1) / 2 = 28/2 = 14

t = (3^4 + 1) / 2 = (81 + 1) / 2 = 82/2 = 41

t = (3^5 + 1) / 2 = (243 + 1) / 2 = 244/2 = 122

t = (3^6 + 1) / 2 = (729 + 1) / 2 = 730/2 = 365

With the formula, you can find any term in the sequence.