Rajasthan Computer Anudeshak Bharti 2026 | Maths Class – संख्या पद्धति (Number System) Part 11

Main ideas / concepts taught

1) Sum of natural numbers (basic formula)

• If the numbers are the natural numbers from 1 to n (consecutive, starting at 1):
  • [ \text{Sum}=\frac{n(n+1)}{2} ]

2) Using the formula to avoid long addition

• In exams, the teacher emphasizes that you should not manually add long sequences like:
  • (1+2+3+\dots+100)
• Instead, substitute into the formula.

3) Handling sums when the sequence does NOT start at 1

Two main approaches are used repeatedly:

Approach A: Add full range then subtract the missing beginning

• For a sum from a to b:

  • [ \text{Sum}(a\text{ to }b)=\text{Sum}(1\text{ to }b)-\text{Sum}(1\text{ to }(a-1)) ]

• Example logic (as shown):

  • For (51+52+\dots+100):
    • Compute (\text{Sum}(1\text{ to }100))
    • Subtract (\text{Sum}(1\text{ to }50))

Approach B: Arithmetic series “shift/pairing endpoints” trick

• For endpoint pairing:

  • [ \text{Sum}=\frac{(\text{first}+\text{last})\times(\text{number of terms})}{2} ]

• The teacher’s wording is messy, but the intent is endpoint pairing.

4) Correct interpretation of keywords in questions

The teacher stresses that many mistakes come from misreading language such as:

• “from x to y” → include both ends: sum of (x) through (y)
• “between x and y” → exclude endpoints:
  • “between 21 and 100” means sum from 22 to 99
• Similar handling for “between 41 to 121”:
  • The teacher interprets “between” as excluding the boundary numbers.

5) Sum of even numbers / odd numbers (special formulas)

Even numbers

• First (n) even numbers are treated as:
  • (2,4,6,\dots)
• The teacher presents:
  • [ \text{Sum(first }n\text{ even)}=n(n+1) ]

Odd numbers

• First (n) odd numbers are treated as:
  • (1,3,5,\dots)
• The teacher presents:
  • [ \text{Sum(first }n\text{ odd)}=n^2 ]

Important caution about “even/odd” language

• If the question says “first n even numbers”, you must treat n as the count of even terms, not as an endpoint value.

6) Sum of odd numbers in a range (odd-sum + subtraction)

• For “odd numbers between A and B”:
  • Compute using the odd-sum formula according to the intended inclusion/exclusion interpretation
  • Then subtract the excluded portion based on the “between” rule.

7) Sums of multiples / multiplication table sums

(a) Sum of the first n multiples of k

• Example: first 50 multiples of 3:
  • Multiples: (3,6,9,\dots,150)

• Used pattern:

  • [ \text{Sum}=3\times(1+2+\dots+50) ]

• Then substitute the natural-sum formula for (1+2+\dots+50).

(b) “Sum of a multiplication table” rule (table up to 10)

• Shortcut stated:

  • [ \text{Sum of the multiplication table of }k\text{ (from }1\text{ to }10)=55k ]

• Reason:

  • (1+2+\dots+10=55)

(c) Extending beyond 10

• For example, “first 120 multiples of 7”:
  • Don’t write the full list
  • Use:
    • [ 7\times(1+2+\dots+120) ]

8) Sum of whole numbers starting from 0 (careful: “whole numbers”)

• The teacher distinguishes:
  • Whole numbers typically start at 0
  • Some confusion occurs when students treat “whole numbers” like natural numbers starting at 1.

9) Homework + session wrap-up

• Homework includes a final practice question (not fully legible).
• Next session will cover:
  • reasoning
  • later topics: sum of squares and sum of cubes.

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Methodology / instruction checklist (as presented)

A) Sum from 1 to n

• Identify the series as: (1,2,3,\dots,n)
• Use:
  • [ \frac{n(n+1)}{2} ]

B) Sum from a to b (when start ≠ 1)

• Identify endpoints:
  • first = (a), last = (b)

• Use either:

  • [ \text{Sum}(a..b)=\text{Sum}(1..b)-\text{Sum}(1..a-1) ]

• Or endpoint pairing:

  • [ \text{Sum}=\frac{(a+b)\times(\text{number of terms})}{2} ]

C) “Between x and y” handling

• If the question says between x and y, set:
  • start (=x+1)
  • end (=y-1)
• Then compute the sum for the adjusted range.

D) Even/odd sums

• If asked first n even numbers:

  • [ n(n+1) ]

• If asked first n odd numbers:

  • [ n^2 ]

• If asked “even/odd in a range”:

  • Convert the range into “first (m)” even/odd counts using inclusion/exclusion
  • Then subtract if needed.

E) Multiples / multiplication table

• If asked “first n multiples of k”:

  • [ k\times(1+2+\dots+n) ]

  • then use ( \frac{n(n+1)}{2}) inside.

  • If asked “sum of k’s multiplication table (1 to 10)”:
  • [ 55k ]

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Speakers / sources featured

• Main instructor (speaker): unnamed (referred to as “Sir” throughout; also addressed in comments with names like “Piyush sir,” but the primary teacher remains the main voice).
• Students / commenters addressed by name:
  • Anjali ji
  • Piyush ji / Piyush sir
  • Darshan ji / Darshan
  • Praveen ji
  • Rihana ji / Rihana Sheikh ji
  • Sheikh ji
  • Ayush ji
  • Chaudhary ji / Chaudhary sahab
  • Kaluram ji
  • Rathore sahab
  • Guru ji