# Next term in sequence [closed]

Came across these 3 sequences to find the next number denoted by '?' in a test yesterday and was unable to select the right answer from the options given

1. 1,6,19,54,151,426,? - Options given are 1415, 1181, 1245, 1201, 1219

2. 20,40,50,110,115,215,? - Options given are 225, 217.67, 220, 218.33, 325

3. $$\frac 14$$ , 1 , $$\frac 3{10}$$, $$\frac 24$$, $$\frac 5{16}$$, $$\frac 37$$, ? - Options given are $$\frac 6{21}$$ , $$\frac 7{22}$$ , $$\frac 4{24}$$ , $$\frac 5{23}$$ , $$\frac 8{26}$$

• Which test? Could you be a bit more specific about the source? Commented Sep 19, 2023 at 12:50
• @Randal'Thor it was an Online Assessment from a Trading firm as the 1st step of their application. Commented Sep 19, 2023 at 15:09
• Looks like this should be three different questions. Commented Sep 27, 2023 at 7:59

0/

Initial thought: Trading firm - Financial - especially consider exponentional behavior

1/

A 'nice increasing' sequence
Looking at the bigger numbers that often show the trend best: I see approximately times 3; also in most answers
When using X[n+1] = 3X[n] the remainders are: 3 (6=3+3), 1 (19-18+1), -3, -11, -27
I did not see an immediate simple progression. Taking differences is often a good idea to try.
This gives the sequence -2, -4, -8, -16; clearly a power of 2.
X[n+1] = 3X[n] - 2^n + C fits for C=5 answer 1219

2/

A sequence with alternating small and big increases -> try odd even subsequences.
20,50,115,? and 40,110,215
I did not recognize anything useful -> try differences.
20,10,60,5,100
Still thinking exponential; note the factors between successive differences: 1/2, 6, 1/12, 20
I immediately recognized the sequence 2,6,12,20.
Difference pattern: 1/(1x2), (2x3), 1/(3x4), (4x5) --> 1/(5x6)
answer 215+100/(5x6) = 218.33. In general, X[n] = X[n-1] + 100 ((n-2)*(n-1)) ^ k, where k = -1 if n is odd, and k = +1 if n is even.

3/

A sequence with alternating small and big numbers -> try odd even subsequences.
1/4, 3/10, 5/16,? and 1, 2/4, 3/7
note the linearly increasing numbers, especially if one writes 1 = 1/1
1,3,5->7 and 4,10,16 ->22 gives as answer 7/22

In part 3, the sequence is: $$\frac{1}{4}$$, $$1$$, $$\frac{3}{10}$$, $$\frac{2}{4}$$, $$\frac{5}{16}$$, $$\frac{3}{7}$$, ?

To solve this I will use two tricks.

My first trick is to

split the sequence into two subsequences. One subsequence will consist of the terms in the odd-numbered positions. The other subsequence will consist of the terms in the even-numbered positions.

The “odd” subsequence is: $$\frac{1}{4}$$, $$\frac{3}{10}$$, $$\frac{5}{16}$$, $$\dots$$

The pattern here is that the numerators are increasing by 2 and the denominators are increasing by 6.

Following this pattern the next odd-numbered term (and the answer to this puzzle) is $$\frac{7}{22}$$.

But to ensure that our answer is reasonable,

let’s look at the “even” subsequence.

My second trick is to rewrite the second term of the original sequence as a fraction.

So the “even” subsequence is: $$\frac{1}{1}$$, $$\frac{2}{4}$$, $$\frac{3}{7}$$, $$\dots$$

The pattern here is that the numerators are increasing by 1 and the denominators are increasing by 3.

So we have a reasonably simple explanation for ALL the terms of the entire original sequence and therefore we have confidence that our expected next term of $$\frac{7}{22}$$ is correct.

Let u denote the sequence. (u[0] = 1, u[1] = 6, ...)

Then the pattern seems to be :

u[n] = u[n-1] * 3 - 2 ^ n + 5


For example :

u[4] = u[3] * 3 - 2 ^ 4 + 5
= 54 * 3 - 16 + 5
= 162 - 11
= 151