# Interleaved series

Interleaved series almost always come up in the IPAT and are one of the easiest to recognize and solve. Lets try to explain this series type with an example:

1, 2, 5, 3, 9, 4, ?

If we split this series into two series –

1, 5, 9 (consists of the 1st 3rd and 5th numbers – all odd numbered terms)

2, 3, 4 (consists of the 2nd 4th and 6th numbers –all even numbered terms)

We see a simple difference based pattern in both series. The first series has a constant difference of 4 while the second series has a constant difference of 1.  The next number in the parent series (which is the 7th term- an odd number) is the next number in the 1st sub series (1,5,9) and hence is equal to 9 + 4 = 13.

The rule of 1, 2, 3

The rule of 1,2,3 gives us a simple framework to approach this type of series. The rule goes as follows: form subseries in the series by first leaving 1 term in between, then 2 terms in between and then 3 terms in between to see which interleaved difference gives a legitimate difference series.

For example, the series above had a subseries gap of 1 term. Here is an example of 2 term subseries

1, 7, 11, 2, 9, 14, 3, 11, 17

Breaking the series into subseries based on 2 term difference

1, 2, 3 (1st, 4th, 7th terms : simple difference series with constant difference of 1)

7, 9, 11 (2nd, 5th, 8th terms : simple difference series with constant difference of 2)

11, 14, 17 (3rd, 6th and 9th terms: simple difference series with constant difference of 3)

Now let’s practice with some IPAT like questions

Q) 7, 6, 14, 12, 21, 18, ?

A] 32

B] 28

C] 14

D] 17

E] 21

Solution

Solution – B] 28. Breaking the series into subseries based on 1 term difference we observe-

7, 14, 21, ? (This is 7×1, 7×2, 7×3)

6, 12, 18 (This is 6×1, 6×2, 6×3)

It is seen that terms in each subseries are multiple of the first number in the subseries. Hence the solution is 7×4 = 28

Q) 2, 3, 4, 4, 9, 16, 16, 81, ?

A] 169

B] 144

C] 256

D] 250

E] 181

Solution

Solution – C] 256. Breaking the series into three subseries with a 2 term (as 1 term does not yield any solution) interleave difference we see.

2, 4, 16

3, 9, 81

4, 16, ?

We observe the next term in each subseries in the square of the previous term in the same subseries.

Hence the solution is 16×16 = 256

Q) 1, 2, 3, 4, 1.1, 3, 6, 4, 1.21, 4.5, 12, 4, 1.331, ?

A] 6.75

B] 1.441

C] 14.11

D] 7.25

E] 8.25

Solution

Solution – A] 6.75. Here is a more complex series, breaking the series into subseries with a 3 term interleave difference we see –

1, 1.1, 1.21, 1.331 ( each next term is formed by multiplying previous term by 1.1)

2, 3, 4.5, ?  (each next term is formed by multiplying previous term by 1.5)

3, 6, 12 (each next term is formed by multiplying previous term by 2)

4, 4, 4 (each next term is formed by multiplying previous term by 1)

Consequently, the next term in the series is 4.5×1.5 = 6.75

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