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Learn about Harmonic Progression, a sequence of numbers where the reciprocals form an arithmetic progression
Understand the formula, properties, and applications of HP in mathematics, physics, and engineering
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A diagram illustrating Harmonic Progression
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A sequence of numbers with a common difference between consecutive terms.
an = a1 + (n-1)d
where:
an = nth term
a1 = first term
n = term number
d = common difference
2, 5, 8, 11, 14 (d = 3)
10, 15, 20, 25, 30 (d = 5)
4, 7, 10, 13, 16 (d = 3)
Common difference (d) remains constant.
Each term is obtained by adding d to the previous term.
Sum of n terms: Sn = n/2 [2a1 + (n-1)d]
First term (a1)
Common difference (d)
nth term (an)
Sum of n terms (Sn)
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A sequence of numbers where each term is obtained by multiplying the previous term by a fixed constant, called the common ratio.
an = a1 × r^(n-1)
where:
an = nth term
a1 = first term
n = term number
r = common ratio
2, 6, 18, 54, 162 (r = 3)
4, 12, 36, 108, 324 (r = 3)
10, 20, 40, 80, 160 (r = 2)
Common ratio (r) remains constant.
Each term is obtained by multiplying the previous term by r.
Sum of n terms: Sn = a1 × (1 - r^n) / (1 - r)
Finite Geometric Progression (FGP)
Infinite Geometric Progression (IGP)
First term (a1)
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nth term (an)
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A diagram illustrating GP
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A picture of a real-world application (e.g., population growth)
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Arithmetic-Geometric Progression (AGP) combines the properties of both Arithmetic Progression (AP) and Geometric Progression (GP)
A sequence where each term is obtained by adding a fixed constant (arithmetic part) and then multiplying by a fixed ratio (geometric part)
an = (a1 + (n-1)d) × r^(n-1)
where:
an = nth term
a1 = first term
n = term number
d = common difference (arithmetic part)
r = common ratio (geometric part)
2, 6, 12, 20, 30 (d = 2, r = 2)
4, 12, 24, 40, 60 (d = 4, r = 2)
10, 25, 40, 55, 70 (d = 5, r = 1.5)
First term (a1)
Common difference (d)
Common ratio (r)
nth term (an)
Arithmetic-Geometric mean
Arithmetic-Geometric Progression Explained
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