Product Series:
A product series is usually a number
series where the terms are obtained by a process of multiplication. Here also,
there can be different types of series. We will look at these through examples.
.
Consider the series 2, 4, 8, 16, 32,
64,………………. .
*2 *2
*2 *2 *2
2, 4, 8, 16, 32, 6
Here, each number in the series is
multiplied by 2 to get the next term. So, the term that comes after 64
is 128. So, each term is multiplied by a fixed number to get the next term.
Similarly we can have a series where we have numbers obtained by dividing the
previous term with a constant number. For example, in the series 64, 32, 16,,
8, ....., each number is obtained by dividing the previous number by 2 (or in
other words, by multiplying the previous term by '/2). So, here, the next term
will be 4 (obtained by dividing 8 with 2). .
Consider the series 4, 20, 80, 240, ....
4, 20, 80, 240
x5 x4 x3
Here, the first term is multiplied by
5 to get the second term; the second term is multiplied by 4 to get the third
term; the third term is multiplied by 3 to get the fourth term. Hence, to get
the fifth term, we have to multiply the fourth term by 2, i.e., the fifth term
is 480. So, each term is multiplied by a decreasing factor (or it could also
be an increasing factor) to get the next term. That is, with whatever number a
particular term is multiplied to get the next term, this latest term is
multiplied by a number different from the previous multiplying factor to get
the next term of the series. All the multiplying factors follow a certain
pattern (normally of increasing or decreasing order). .
Consider the series 2, 6, 12, 20, 30,…..
2, 6, 12, 20, 30
+4 +6 +8 +10
This can be looked at a series of
increasing differences. The differences of consecutive pairs of terms are 4
(between 2 and 6), 6 (between 6 and 12), 8 (between 12 and 20), 10 (between 20
and 30) and so on. Hence, the difference between 30 and the next term should be
12 and so the next term will be 42. But this series can also be looked at as a
product series. .
2, 6, 12, 20, 30
1 x 2 2x3 3x4 4 x 5 5 x 6
The first term is the product of 1 and
2; the second term is the product of 2 and 3; the third term is the product of
3 and 4; the fourth term is the product of 4 and 5; the fifth term is the
product of 5 and 6. Hence, the next term will be the product of 6 and 7 that is
42. .
