In the last article, I had discussed various types of series. Today I will discuss how to approach a series problem once you have identified what kind it is. Let me be honest. There is no algorithm for solving number series questions appearing in quantitative section of any competitive exam because each question is different and there are innumerable ways in which a series can be generated. Cracking a series question involves a lot of practice and the intuition that comes with practice.
Series Questions – Samples
- 2, 4, 8, 16, 32, 64
- 729, 512, 343, 216, 125, 64
- 1, 1, 2, 4, 6, 10, 16, 26, 42, 68… (Fibonacci Series)
- 4, 9, 6, 18, 9, 27, 13, 36
I will today broadly discuss steps how to build this intuition so that you can solve series questions easily and quickly.
Step 1: Screening
First check the series by giving a cursory look at it. Many a times a careful first look may be enough to tell the next term. Have a look at these examples:
- 3, -9, 27, -81, 243, ?
- 1, 3, 7, 13, 21, 31, ?
- 1, 3, 6, 10, 15, 21, ?
- 2, 4, 8, 16, 32, 64, ?
- Each successive term is multiplied by -3. Next term will be 243 X -3= -729.
- The series is +2, +4, +6, +8, +10, +12. So next term will be 31 + 12 = 43.
- The series is +2, +3, +4, +5,+6, +7. Next term will be 21 + 7 = 28.
- Each term is multiplied by 2. Next term will be 64 X2 = 128.
Step 2: check the pattern: increasing/ decreasing/ alternating
In case you fail to decipher the rule of the series by just preliminary series, try to understand the trend of the series. Look at the pattern. Is it increasing or decreasing? Is it following an alternating pattern?
- 1, 4, 9, 16, 25, 36, 49.
- 4, -8, 16, -32, 64, -128.
- 729, 512, 343, 216, 125, 64.
- 5, 10, 13, 26, 29, 58.
Clearly, i and iv are following an increasing pattern. ii is following an alternating pattern while iii is a decreasing pattern series .
Step 3: if the series is increasing or decreasing, find the rate of increase or decrease
Start with the first term and move onto the next. Gauge whether the series proceeds arithmetically or geometrically or alternately. You have to feel whether this rise or fall is slow or fast. In an arithmetic progression, the increase or decrease of terms is by virtue of addition or subtraction. So the rise or fall will appear slow. In contrast, in a geometric progression the increase or decrease of terms is by virtue of multiplication or division. So naturally the rise or fall will be very fast. If the geometric progression involves squaring or cubing, this rise or fall will be even sharper. I hope you understand how you need to feel the rate of increase or decrease.
In the above examples now it is clear that 1, 4, 9, 16, 25, 36, 49,… is an arithmetic type of series. The trend being +3, +5, +7, +9 and so on. Clearly, 729, 512, 343, 216, 125, 64,… is a geometric type of series involving cubes of 9, 8, 7, 6, 5, 4 and so on.
Now look at 1, 5, 14, 30, 55, 91. It can easily be figured out that the rise is rather sharp. So it is a geometric progression. On trial you will see that just successive multiplication is not involved here. Checking for addition of squared numbers or cubed numbers we see that this is actually 12, 12+22, 12+22+32 and so on.
Basically, a geometric increase or decrease can take place in two ways –
- Multiplication or division by terms
- Addition or subtraction of squared or cubed terms
Obvious question is how does one differentiate between the two?
Look at the trend of the increase. If the increase is because of addition of squared or cubed numbers, the increase will be very sharp initially but not very sharp in the later terms say 5th or 6th term onwards.
Watch the earlier example: 1, 5, 14, 30, 55, 91.
1, 1X5=5, 5X2.8=14, 14X2.1428=30, 30X1.8333=55, 55X1.6545= 91.
Notice how the multiplication factor is gradually decreasing from 5 in the initial step to 1.6545 in the last step. Hence, where the rise is very sharp initially but gradually slows down, it should strike in your mind that you have to look for a pattern involving addition of squared or cubed numbers. You needn’t do the above calculations of multiplication factor, just try to build the intuition of understanding the rate of increase or decrease.
Consider 4, 5, 12, 39, 121, 610. This series also rises very sharply. Hence this must be a geometric progression. However the rate of increase does not slow down in later terms. In fact it picks up as the series progresses. Therefore we can conclude that the series must be of the first kind i.e. formed by multiplication. A little more exercise will tell us that the series is : x1+1, x2+2, x3+3, x4+4, x5+5 and so on.
Now let me show you an example of alternating increase. Two possibilities will exist here:
- Two different series may be intermixed
- Two different kind of operations may be being performed on successive terms
Consider the series 1, 4, 5, 9, 14, 20, 30, 43. This series increases gradually but the increase is rather haphazard. Actually it is a mix of two series!
1, 5, 14, 30, 55, 91 which is a series: 1, 1+22, 1+22+32, 1+22+32+42 ; and 4, 9, 20, 43, 90 which is a series: x2+1, x2+2, x2+3, x2+4.
Again, look at 3, 13, 18, 76, 81, 409, 414. Here two different operations are being performed alternately: the first operation is that of multiplication by 3, 4 and 5 successively and adding a constant number 4 and the second operation is adding a constant number 5. Hence the series is x3+4, +5, x4+4, +5, x5+4, +5.
Step 4: Check if the series is neither increasing nor decreasing but alternating
For an alternating series also you should check whether it is a mix of two series or two different operations are being performed alternately.
For example, 4, 9, 6, 18, 9, 27, 13, 36. It is a mix of two series 4, 6, 9, 13 and 9, 18, 27, 36.
In 200, 600, 1200, 1600, 3200, 3600, 7200 two operations are going on; one addition of 400 and second multiplication by 2.
With practice these steps can help you build the intuition required to solve problems related to series. Did you ever face any difficulty in solving number series questions?