Arithmetic, Geometric & Harmonic Progressions (nth term, sum, and more)
An HP is the reciprocal of an AP. Enter a and d of the underlying AP. nth term of HP = 1/(a+(n-1)d).
Quick Answer – What are Sequences & Series?
A sequence is an ordered list of numbers; a series is the sum of its terms. An Arithmetic Progression (AP) has a constant difference between terms. A Geometric Progression (GP) has a constant ratio. A Harmonic Progression (HP) is formed by taking reciprocals of an AP. Formulas for nth term and sum are essential for JEE and NCERT.
Key Formulas
AP: Tn = a + (n−1)d, Sn = n/2 [2a + (n−1)d]
GP: Tn = a rn−1, Sn = a(rn−1)/(r−1) for r>1, S∞ = a/(1−r) for |r|<1
HP: nth term = 1/(a + (n−1)d), where a and d come from the related AP
Solved Examples
Example 1: AP – 5th term of 2,5,8,...
a=2, d=3, n=5 → T5 = 2 + (5-1)×3 = 2+12 = 14
Example 2: GP – Sum of first 4 terms of 3,6,12,...
a=3, r=2, n=4 → S4 = 3(2⁴-1)/(2-1) = 3(16-1) = 45
Example 3: HP – 4th term of 1/2, 1/5, 1/8,...
Reciprocal AP: 2,5,8,... → a=2,d=3,n=4 → T4 of HP = 1/(2+3×3) = 1/11 ≈ 0.0909
Frequently Asked Questions
Q1. Can I use decimal values for a, d, r?
Yes, any real numbers work. For GP, r can be negative or a fraction. The infinite sum only works if |r| < 1.
Q2. What if n is not an integer?
n represents the term position and should be a positive integer. The calculator validates this.
Q3. How do I enter a negative common difference or ratio?
Simply type a minus sign, e.g., −2. The formulas handle negative values correctly.
Q4. Can I get the sum of an infinite GP?
Yes, select "Infinite sum" in the GP tab. The calculator checks |r|<1 first.
Q5. Why does HP use AP parameters?
Because a harmonic progression is defined by the reciprocals of an arithmetic progression. To find an HP term, you find the corresponding AP term and take its reciprocal.