Tricks To solve AP, GP and HP Questions Quickly
Last Updated : 08 Nov, 2024
The topic of progressions—Arithmetic Progression (AP), Geometric Progression (GP), and Harmonic Progression (HP)—plays a crucial role in various mathematical concepts and competitive exams. While these progressions may seem complex at first glance, mastering a few simple tricks and methods can make them much easier to understand and apply.
A progression is a sequence of numbers where the relationship between consecutive terms follows a specific pattern. AP, GP, and HP each have their unique rules and formulas that govern how their terms are related. These progressions are widely used in various fields, including algebra, calculus, and number theory, and are often tested in competitive exams to evaluate problem-solving skills.
In an arithmetic sequence, the difference between any term and the next term is constant. Specifically, a sequence a1, a2, ..., an is called an arithmetic sequence or arithmetic progression, if the difference between consecutive terms (d = an+1 - an) where d is a constant known as the common difference.
Example of Arithmetic Progression is as follows:
a, a+d, a+2d, a+3d, ...
Where,
- a is the first term of the Arithmetic Sequence.
- d is a common difference between any two consecutive terms of the Arithmetic Sequence.
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Tricks and Formulas for Arithmetic Progression (AP)
1. Finding the nth Term Quickly
To find the nth term, simply plug the values of a, n, and d into the formula:
an = a1 + (n − 1) × d
2. The Sum of n Terms Shortcut
- Using the standard sum formula:
The sum of the first n terms in an AP can be calculated using:
Sn = n/2 × (2a1 + (n − 1) × d)
- Alternatively, using the first and last term:
If the last term an is known, use:
Sn = n/2 × (a1 + an)
3. Condition for Three Numbers in AP
If A, B, and C are in arithmetic progression, then:
2B = A + C
4. Relation Between Terms:
- a1 + an = a2 + an−1
- 2an = an+ k +an−k
5. Total Number of Terms:
The total number of terms can be calculated as:
(a1 + an)/d +1
where
- ai is the first term.
- an is the last term.
- d is the common difference.
Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio.
The general form of GP is as follows:
a, ar, ar2, ar3,…
Where:
- a is the first term,
- r is the common ratio.
Tricks and Formulas for Geometric Progression (GP)
1. Common Ratio
Tn and Tn-1 are consecutive terms of the GP.
r = Tn/ Tn-1
2. Finding the nth Term Quickly
To find the nth term, simply plug the values of a, n, and r into the formula:
Tn = arn-1
3. The sum of n Terms
For (r > 1):
Sn is the sum of the first n terms for r > 1.
Sn = a[(rn – 1)/(r – 1)]
For (r < 1):
Sn is the sum of the first n terms, for r < 1.
Sn = a[(1 – rn)/(1 – r)]
4. Relationship between Terms:
The product of the first and the last term equals the product of the second and the second-last term, and so on:
a1⋅an = a2⋅an − 1 = ak⋅an−k + 1
5. Product of First and Last Terms:
For terms a1, a2, and an in a GP,
We have the relationship:
b2 = ac
6. Geometric Mean (GM):
The geometric mean (GM) of terms a1, a2, … ,an in a GP is given by
GM = \sqrt[n]{a1 * a2 * ... * an}
A Harmonic Progression (HP), or Harmonic Sequence, is a sequence of real numbers derived by taking the reciprocals of the terms in an Arithmetic Progression (excluding 0). In this sequence, each term is the Harmonic Mean of its two neighboring terms.
Example of Harmonic Progression is as follows:
1/a, 1/(a + d), 1/(a + 2d), 1/(a + 3d), ...
Where
- “a” represents the initial term of the Harmonic Progression (H.P.)
- “d” stands for the common difference between successive terms,
- “n” denotes the total number of terms in the Harmonic Progression (H.P.)
Tricks and Formulas for Harmonic Progression (HP)
1. Finding the nth Term Quickly
The nth term of a Harmonic Progression (HP) is given by the reciprocal of the corresponding term in the Arithmetic Progression (AP). So, the n-th term of HP can be written as:
an = 1/(a + (n − 1)d)
2. The sum of n Terms for Harmonic Progression
To find the Sum of n terms in a Harmonic Progression (Sn) for the sequence 1/a, 1/a + d, 1/a + 2d, . . ., 1/a + (n−1)d, the formula is:
S_n \approx \frac{1}{d} \cdot \ln\left(\frac{2a + (2n - 1)d}{2a - d}\right)
where,
- “a” denotes the first term of the Harmonic Progression (H.P.)
- “d” represents the common difference in the Harmonic Progression (H.P.)
- “ln” stands for the natural logarithm.
Note: This formula only gives the approximate required value.
3. Harmonic Mean
The Harmonic Mean of two numbers a and b is defined as:
Harmonic Mean = 2ab/(a + b)
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