Number Series

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🔢 Number Sequence Basics

Find number patterns and calculate the next number!

1️⃣ Number sequence types

Type	example	style
addition sequence	2, 5, 8, 11, 14	+3
subtraction sequence	20, 17, 14, 11, 8	-3
Multiplication sequence	2, 4, 8, 16, 32	×2
Dividing order	64, 32, 16, 8, 4	÷2
class order	1, 4, 9, 16, 25	n²
cubic order	1, 8, 27, 64, 125	n³

2️⃣ difference order

The first difference is:Difference of consecutive numbers
Second Difference:Difference of differences

Example:1, 4, 9, 16, 25

Difference: 3, 5, 7, 9 (+2 styles)

3️⃣ Major classes

1²=1	2²=4	3²=9	4²=16	5²=25
6²=36	7²=49	8²=64	9²=81	10²=100
11²=121	12²=144	13²=169	14²=196	15²=225

4️⃣ Major weights

1³=1	2³=8	3³=27	4³=64	5³=125
6³=216	7³=343	8³=512	9³=729	10³=1000

5️⃣ Mixed Rows

Dynamic addition:+1, +2, +3, +4...
Dynamic Multiplication:×1, ×2, ×3...
Dual Style:Different style in single/double spaces
Fibonacci:The sum of the previous two

📐 Number sequence formulas

🔢 Basic formulas

sequence	nth element
Additive Order (AP)	a + (n-1)d
Multiplicative Sequence (GP)	a × r^(n-1)
class order	n²
cubic order	n³
Fibonacci	Fₙ = Fₙ₋₁ + Fₙ₋₂

📊 Summation formulas

1+2+3+...+n	n(n+1)/2
1²+2²+3²+...+n²	n(n+1)(2n+1)/6
1³+2³+3³+...+n³	[n(n+1)/2]²

🎯 AP formulas

nth element:aₙ = a + (n-1)d
Sum:Sₙ = n/2 [2a + (n-1)d]
Middle element:(first + last)/2

🔄 GP formulas

nth element:aₙ = a × r^(n-1)
Sum:Sₙ = a(rⁿ-1)/(r-1) [r≠1]

📚 Solved Questions – 10

Question 1: Addition sequence

2, 5, 8, 11, ? What's next?

Solution:Difference = +3
11 + 3 =14

Question 2: Multiplication order

3, 6, 12, 24, ? What's next?

Solution:Style = ×2
24 × 2 =48

Question 3: Class Order

1, 4, 9, 16, 25, ? What's next?

Solution:n² array (1², 2², 3², 4², 5², 6²)
6² =36

Question 4: Cubic order

1, 8, 27, 64, ? What's next?

Solution:n³ array (1³, 2³, 3³, 4³, 5³)
5³ =125

Question 5: Fibonacci

1, 1, 2, 3, 5, 8, ? What's next?

Solution:The sum of the previous two
5 + 8 =13

Question 6: Dynamic addition

2, 3, 5, 8, 12, ? What's next?

Solution:Difference = +1, +2, +3, +4, +5
12 + 5 =17

Question 7: n²+1 sequence

2, 5, 10, 17, 26, ? What's next?

Solution:n²+1 (1²+1, 2²+1, 3²+1...)
6²+1 = 36+1 =37

Question 8: Wrong no

2, 4, 8, 16, 30, 64 - Which is the wrong number?

Solution:×2 array (2,4,8,16,32,64)
Error =30(correct = 32)

Question 9: Dual style

1, 2, 4, 5, 7, 8, 10, ? What's next?

Solution:+1, +2, +1, +2... style
10 + 1 =11

Question 10: Summation

1+2+3+...+10 = ?

Solution:n(n+1)/2 = 10×11/2 =55

⚡ Speed shortcuts

🎯 Way to find style

First see the difference (+/-)
If the difference is equal → AP
If the ratio is equal → GP
Check for class/weight numbers

🔢 To remember

Addition of 1-10	55
Addition of 1-20	210
Addition of 1-100	5050

📊 Square-cube reminder

Class:1,4,9,16,25,36,49,64,81,100
Weight:1,8,27,64,125,216,343,512,729,1000

⚡ Speed formulas

1+2+...+n= n(n+1)/2
Single number addition= n²
Double number addition= n(n+1)

💡 Exam Tips

Calculate the difference first
Exponents of 2: 2,4,8,16,32,64,128,256
Prime Numbers: 2,3,5,7,11,13,17,19,23
Fibonacci: 1,1,2,3,5,8,13,21,34,55

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