Arithmetic, Geometric Progression and Harmonic Progression for JEE

Before we venture into the meaning of terms like arithmetic, geometric progressions, and harmonic progressions, we must first understand what a passage is. In Mathematics, as well as in real life, an individual comes across several examples of progression. A progression is nothing but a series that exhibits a particular pattern. In the case of a number system, a passage is a series or sequence of numbers that either have a common difference or a standard ratio, or whose common values exhibit some pattern. 

Arithmetic progressions are a series of numbers where the difference between the first and the second term, the second and the third term, and so on is the same. The difference between two consecutive terms of the series is termed a ‘common difference.’ A geometric progression is slightly different from an arithmetic progression. Here, the values in the progression are multiplied by the same factor, which is termed the common difference. A harmonic progression is one where the reciprocals of the terms constituting the sequence are in AP. Read on to learn more about the AP, GP, and HP formulas, examples, and their differences. 

What Is Arithmetic Progression?

As discussed above,  an arithmetic progression is any series of numbers, fractions, or values where two consecutive terms have the same difference. The difference between any two adjacent terms is known as a common difference, which is usually represented by d. If we have the first term of the series, the common difference, and the number of terms in that particular sequence, we can find out the values of the other terms. 

Formulas Related to an Arithmetic Progression

Let a₁ , a₂, a₃, a₄,…. Be an arithmetic progression.

The first term of the series is a₁ and the common difference is represented by d.

Therefore, the common difference d = a₂ - a₁ 

If there are n terms in the series then the n^th term will be given by $a_{n}=a_{1}+(n-1) d$

The sum of all the terms of an arithmetic series with n terms is given by $S=\dfrac{n}{2}\left(a_{1}+a_{n}\right)$. The more general formula for finding the value of any number of terms, say through terms m ( any term before the n^th term) to n, is $S=\dfrac{n-m+1}{2}\left(a_{m}+a_{n}\right)$.

Arithmetic Progression Examples   

5,10,15,20,25,30.

In the above-given series, the first term a₁ = 5, the common difference d = ( 10-5)=5. The total number of terms, n = 6

Therefore the sum of the terms $S=\dfrac{n}{2}\left(a_{1}+a_{n}\right)=\dfrac{6}{2}(5+30)=105$

What Is Geometric Progression?

Just like how the terms of an arithmetic progression are connected by the common difference, a sequence where any two adjacent terms are related by the common ratio is conventionally represented by r. By dividing any term by the next consecutive term, we can derive the common ratio.

Formulas Related to a Geometric Progression

Let a₁, a₂, a₃, a₄,…. be a geometric progression.

Let the total number of terms in the geometric series be n.

Therefore, the common ratio is $r=\dfrac{a_{2}}{a_{1}}$ ( here a₁  is the first term of the series)

The sum of all the terms of the geometric progression when there are n number of terms will be given by the formula, $S_{n}=a\left(\dfrac{r^{n}-1}{r-1}\right)$

Here, r does not equal 1, but r is greater than 1.

The nth term of the series is obtained from the formula $a_{n}=a_{1} r^{n-1}$.

Geometric Progression Example

$1, \dfrac{1}{2}, \dfrac{1}{4}, \dfrac{1}{8}$

Here, in the given example, the first term of the series a₁  = 1.

The common ratio $r=\dfrac{1}{2}$ and the number of terms n = 4

Therefore, the sum of all the terms of the geometric progression series will be $=\dfrac{1\left(1-\left(\dfrac{1}{2}\right)^{4}\right)}{1-\dfrac{1}{2}}=\dfrac{15}{8}$

What Is Harmonic Progression?

A harmonic progression is probably the most confusing of the three. At first glance, the series might seem like a bunch of arbitrary numbers with no distinct pattern. However, that is definitely not the case. In the case of a harmonic progression, the reciprocals of all the terms of the series are in an arithmetic progression. 

Harmonic Progression Formula

$H . P .=\dfrac{1}{a+(r-1) d}$

Here,

“a” is the first term of A.P

“d” is the common difference

“n” is the number of terms in A.P

Harmonic Progression Example:

If the 10th and 20th terms of HP are 20 and 40, calculate the 100th term of HP.

Solution: The following is the matching A.P to the supplied H.P,

20th term of H.P.=$ a+19d= \dfrac{1}{40}$…(1)

10th term of A.P.= $a+9d= \dfrac{1}{20}$…(2)

For equation (1) and (2)

After the calculation of,

$a=\dfrac{ 29}{400}$, $d=\dfrac{-1}{400}$

We should write the expression in the form, to find the 100th term,

$a+99d= \dfrac{29}{400}+99\left(\dfrac{-1}{400}\right)$

$\begin{align} &\Rightarrow a+99 d=\dfrac{29}{400}-\dfrac{99}{400} \\ &\Rightarrow a+99 d=\dfrac{29-99}{400} \\ &\Rightarrow a+99 d=\dfrac{-70}{400} \end{align}$

11th term of H.P.= $\dfrac{1}{100^{\text {th }} \text { term of A.P. }}$

11th term of H.P.= $\dfrac{400}{-70}=\dfrac{-40}{7}$

The 100th term of HP is $\dfrac{-40}{7}$.

Difference Between Arithmetic Progression and Geometric Progression

Two very basic arithmetic operations include addition and multiplication. Multiplication is nothing more than repeated addition. A geometric progression includes terms that are products of the terms right before them and the common ratio. An arithmetic progression includes terms where one term is nothing but the sum of the terms right before it and the common difference. Therefore, it might seem like a valid argument that a geometric progression is in reality an arithmetic progression minus certain terms, or rather an arithmetic progression is a geometric progression with certain terms being absent. 

However, there are some key differences between arithmetic progression and geometric progression. The following table gives us the main points of difference between the two:

Conclusion

A progression in Mathematics is a series of numbers that exhibits a particular pattern. An arithmetic progression is a series where the difference between any two adjacent terms is one and the same. Similarly, a geometric progression is one where any two consecutive terms are related by the common ratio. A harmonic progression is such a sequence where the reciprocals of all the terms of the progression are in an arithmetic progression. The current topic, the progression unit in the board exam, has intermediate levels. The theoretical and numerical aspects of this material are crucial from an exam standpoint because the board test is subjective. The score advantage section also provides some numerical information. The progression unit has a moderate weighting because it only contains a small amount of mathematics.