Arithmetic & Geometric Progression

A&G Progression

What is Arithmetic?

An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant.

For example, the sequence 1, 2, 3, 4, ... is an arithmetic progression with common difference 1.
Second example: the sequence 3, 5, 7, 9, 11,... is an arithmetic progression
with common difference 2.
Third example: the sequence 20, 10, 0, -10, -20, -30, ... is an arithmetic progression
with common difference -10.

Rule for the nth term (1st)

 with this formula you have to identify what is an, a1 and so are the rest. below is the meaning for every each value:

a_n = Term itself

a₁ = First sequence / term

d = common difference

n = # of the term

Example & questions

The first term of an arithmetic sequence is equal to 6 and the common difference is equal to 3. Find a formula for the nth term and the value of the 50th term.

a_n = 6 + (3 (50 -1)   

a_n = 6 + 147

_Ans = 153

As you can see on the provided question, it already give the hint which are for the first term is 6, the common difference is 3 and whereas the term itself  is the 50th.

2nd Formula to find AP

 Example & questions

Determine the sum of the first 25 terms of Arithmetic Series?:

S₂₅ = n / 2 (2a₁ + ( n – 1 ) d)

 a = 20, d = -2, n = 25

S₂₅ = n / 2 (2a₁ + ( n – 1 ) d)

S₂₅ = 25 / 2 (2(20) + (25 -1)(-2)

Ans =  -100

This is the same thing like the first formula but the only difference is to find  the 'S'(sum). In addition, you have to find the common difference first and select which one is n.

 What is Geometric Progression?

In mathematics, a geometric progression (also inaccurately known as a geometric series) is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence.

 

a_n = Term itself

a₁ = First sequence / term

r = common ratio

n = # of the term

 

GP 1st Formula

  

Example & questions

Find the 7th term?

 2, 6, 18, 54, ........

a_n = a1(r)^n-1                                        

  

a₇ = 2(3)^7-1

 a_{7 =} 2(3)⁶

 a_{7 =} 2(729)

  

_{Ans = 1458}

  First all you need to do is find the common ration. To find the common ratio you must use Multiply and Divide only. to find it for example you have to read the question first, then once you know what number to be multiply with the number of 2 then you will find the common ratio (for the next term). whereas using divide sign, you just divide with the previous term like the 2nd term divide by 1st term.

GP 2nd Formula

Example & questions

Find the sum of the first 8 term?

-5, 15, -45, 135, _, _ , _, _.

a₁ = -5

 r = -3

 n = 8

  

This is the same thing like the 1st formula, all you need to do is state what is the first term, common ratio and number of the term.

 after that, you may begin use the 2nd fornula which as you can see on the picture above.

S_n =  a₁ (1- rⁿ) / 1 - r

S₈ = -5  (1- (3)⁸) / 1 -(-3)

Ans = 8,200