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Besides Arithmetic Progression(AP), students will also be introduced to another form of progression that is known as Geometric Progression(GP). Unlike the AP that is comprised of a sequence of numbers (ascending or descendingly) such that each number is obtained by way of adding the preceeding number by a constant (common difference or d), GP is actually a sequence of numbers such that each number is obtained by way of multiplying the preceeding number by a constant (common ratio or r).

Example of GP are 3, 9, 27, 81, …… 1/4, 1/8, 1/16, 1/32, 1/64,…..

If we study the number sequence of 1, 3, 9, 27, 81, ……, the constant involved is 3 which can be obtained by way of dividing the number term of n by (n-1) ; 3/1, 9/3 or 27/9 and so on….

From the info above, a = 1 and r = 3

Take note that T₁ or a can never be 0 and r can never be 1 or 0

T₁ = 1 = a

T₂ = 3 = 1 x 3 = ar

T₃ = 9 = 1 x 3 x 3 = ar^2 = ar^(3-1)

T₄ = 27 = 1 x 3 x 3 x 3 = ar^3 = ar^(4-1)

T₅ = 81 = 1 x 3 x 3 x 3 x 3 = ar^4 = ar^(5-1)

Based on the steps and calculation shown above, we can summarise the method by finding the term of n = ar^(n-1)

Please take note that 2^3 = 2³

GP1

Now let us study the method of calculating the Sum of GP term. By taking the same example of 1, 3, 9, 27, 81, ……

S₁ = 1 = a —> T₁

S₂ = 1 + 3 = 4 —> T₁ + T₂ = a + ar

S₃ = 1 + 3 + 9 = 13 —> T₁ + T₂ + T₃  = a + ar + ar²

S₄ = 1 + 3 + 9 + 27 = 40 —> T₁ + T₂ + T₃ + T₄  = a + ar + ar² + ar³

S₅ = 1 + 3 + 9 + 27 + 81 = 121 —> T₁ + T₂ + T₃ + T₄ + T₅  = a + ar + ar² + ar³ + ar⁴

By taking  S₄ as our example, we can summarise the method in finding the sum of 4th term as follows :

a + ar + ar² + ar³ = a (1 + r + r² + r³) and in order to further summarise the equation, we multiply both the numerator and denominator by (r – 1)

a (1 + r + r² + r³)(r – 1)/(r – 1) = a ( r + r² + r³ + r⁴ – 1 – r – r² – r³)/(r – 1)

                                               = a ( r⁴ – 1 )/ (r – 1)

As such, our summarised formulae for Sum of GP term shall be as follows :

GP2

Further examples on GP will be posted soon….

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