Combination Series:
A number series which has more than
one type of (arithmetic) operation performed or more than one series combined
together is a combination series. The series that are combined can be two
series of the same type or could be different types of series as described
above. Let us look at some examples. .
First let us look at those series
which are formed by more than one arithmetic operation performed on the terms
to get the subsequent terms.
Consider the series: 2, 6, 10, 3, 9,
13, 4, 12, Here, the first term 2 is multiplied by 3 to get the second term,
and 4 is added to get the third term. The next term is 3 (one more than the
first term 2) and it is multiplied by 3 to get 9 (which is the next term) and
then 4 is added to get the next term 13. The next term 4 (which is one more
than 3) which is multiplied with 3 to get 12. Then 4 is added to this to get
the next number 16.
Consider the series: 1, 2, 6, 21,
88, Here, we can observe that 88 is close to 4 times 21. It is in fact 21 x 4
+ 4. So, if we now look at the previous term 21, it is related to the previous
term 6 as 6 x 3 + 3. Now we get the general pattern: to get any term, multiply
the previous term with k and then add k where k is a natural number with values
in increasing order from 1.So, to get the second term, the first term has to be
multiplied with 1 and then 1 is added. To get the third term, the second term
is multiplied with 2 and then 2 is added and so on. Hence, after 88, the next
term is 88 x 5 + 5, i.e., 445.
Now, let us look at a series that is formed
by combining two (or more) different series. The two (or more) series can be of
the same type or of different types described above. .
Consider the series: 8, 12, 9, 13, 10, 14,
... . . Here the 1', 3', 5th, ... terms which are 8, 9, 10, ..... form one
series whereas the 2", 4th, 6th, etc. terms which are 12, 13, 14 form
another series. Here, both series that are being combined are two simple
constant difference series. Therefore the missing number will be the next term
of the first series 8, 9, 10, ... which is equal to 11. .
Consider the series: 0, 7, 2, 17, 6, 31, 12,
49, 20, .... Here, the series consisting of l', 3rd, 5th, terms (i.e., the
series consisting of the odd terms) which is 0, 2, 6, 12, 20, ... is combined
with another series consisting of 2", 4th, 6th, ... terms (i.e., the
series consisting of the even terms) which is 7, 17, 31, 49, ... . . The first
series has the differences in increasing order 2, 4, 6, 8, 10 and so on. The
second series also has the difference in increasing order 10, 14, 18, Since,
the last term 20 belongs to the first series, a number from the second series
should follow next. The next term of the second series will be obtained by
adding 22 to 49, that is 71.
Consider the series: 1, 1, 2, 4, 3, 9, 4,
16, ...... . . Here, one series consisting of odd terms, which is 1, 2, 3, 4,
...., is combined with the series of even terms which is 1, 4, 9, 16, ..... The
first series is a series of natural numbers. The second series is the squares
of natural numbers. Hence, the next term is 5. .
Consider the series: 1, 1, 4, 8, 9, 27,
Here, the series of squares of natural numbers is combined with the series of
cubes of natural numbers. The next term in the series will be 4.
Consider the series: 2, 4, 5, 9, 9, 16, 14,
? , 20, Here, we have to find out the term that should come in place of the
question mark. The odd terms form one series 2, 5, 9, 14, 20, ….
where
the difference is increasing the differences are 3, 4, 5, series is combined
with the series of even terms 4, 9, 16, her c the terms are , squares of
numbers 2, 3, 4,…..hence the terms that should come in place of the question
mark is the next terms of the second series which is 5', i.e., 25.
