Q1. Can you explain why
\[1+2+1, 1+2+3+2+1, \dots\]
give square numbers?
Solution:
These numbers form a pattern of increasing and then decreasing.
\[1+2+3+\dots+n+\dots+3+2+1 = n^2\]
👉 Hence, the result is always a square number.
Q2. Find the value of
\[1+2+3+\dots+99+100+99+\dots+2+1
\]
Solution:
This follows the same pattern as Q1.\[
= 100^2 = 10000
\]
Q3. What sequence do you get when you add all 1’s?
(i) Adding up:
\[
1,1+1=2,3,4,5,\dots
\]
👉 Sequence: 1, 2, 3, 4, 5, … (Counting numbers)
(ii) Adding up and down:
\[
1,1+1+1=3,1+1+1+1+1=5
\]
👉 Sequence: 1, 3, 5, 7, … (Odd numbers)
Q4. What sequence do you get when adding counting numbers?
\[1 = 1
\]
\[
1+2 = 3
\]
\[
1+2+3 = 6
\]
\[
1+2+3+4 = 10
\]
👉 Sequence: 1, 3, 6, 10, 15, … (Triangular numbers)
Q5. What happens when you add consecutive triangular numbers?
\[1+3 = 4
\]
\[
3+6 = 9
\]
\[
6+10 = 16
\]
\[
10+15 = 25
\]
👉 Sequence: 4, 9, 16, 25, … (Square numbers)
👉 Because:
\[
T_n + T_{n+1} = (n+1)^2
\]
Q6. What happens when you add powers of 2?
\[1 = 1
\]
\[
1+2 = 3
\]
\[
1+2+4 = 7
\]
\[
1+2+4+8 = 15
\]
👉 Pattern:
\[
= 2^n - 1
\]
Now adding 1:
\[
2,4,8,16,\dots
\]
👉 Sequence: Powers of 2
Q7. What happens when you multiply triangular numbers by 6 and add 1?
Triangular numbers:
\[
1,3,6,10,15,\dots
\]
Now calculate:
\[
6 \times 1 + 1 = 7
\]
\[
6 \times 3 + 1 = 19
\]
\[
6 \times 6 + 1 = 37
\]
\[
6 \times 10 + 1 = 61
\]
\[
6 \times 15 + 1 = 91
\]
👉 Sequence obtained:
\[
7,19,37,61,91,\dots
\]
👉 This is the sequence of centered hexagonal numbers.
Q8. What happens when you add hexagonal numbers?
Given pattern:
\[
1,1+7=8, 1+7+19=27, 1+7+19+37=64
\]
👉 So we get:
\[
1,8,27,64,\dots
\]
👉 These are:
\[
1^3,2^3,3^3,4^3,\dots
\]
👉 Sequence obtained: Cubes
\[= n^3
\]
Q9. Find your own patterns
Some simple patterns:
(i) Odd numbers sum → Squares
\[
1 = 1^2
\]
\[
1+3 = 4 = 2^2
\]
\[
1+3+5 = 9 = 3^2
\]
(ii) Even numbers sum
\[
2 = 2
\]
\[
2+4 = 6
\]
\[
2+4+6 = 12
\]
👉 Pattern increases by adding even numbers.
(iii) Powers of 2
\[
1,;2,;4,;8,;16,\dots
\]
👉 Each number ×2
(iv) Triangular relation
\[
T_n + T_{n+1} = \text{Square number}
\]
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