Averages of Number Series
Consecutive numbers का average
Learning Objective
Use symmetry: for evenly spaced numbers, average = middle value.
🎯 Learning Objective
Use symmetry: for evenly spaced numbers, average = middle value.
💡 Concept
- Evenly spaced numbers → average = (first + last) ÷ 2 = middle term
- First n natural numbers → average = (n+1)/2
- First n even numbers → average = n+1
- First n odd numbers → average = n
- Consecutive numbers with odd count → average is the middle number
🧮 Key Formulas
Evenly spaced: Avg = (first + last)/2
>
1..n → (n+1)/2 | even → n+1 | odd → n
✏️ Easy Example
Q. Find the average of 21, 23, 25, 27, 29.
- Evenly spaced, middle term = 25
Answer: 25
🇮🇳 Real-Life Example
Seat numbers 41–45 in a train coach: the 'average seat' is 43, the middle one — no addition needed.
📝 Exam-Level Example
Q. The average of 5 consecutive numbers is 18. Find the largest.
- Middle number = 18
- Numbers: 16, 17, 18, 19, 20
Answer: 20
📝 Exam-Level Example
Q. Average of first 50 even numbers?
- Formula: n + 1 = 50 + 1
Answer: 51
🪄 Memory Trick
Consecutive-number questions: place the average in the MIDDLE and count outward.
⚠️ Common Mistakes
- ❌ Adding long series manually
- ❌ Using (n+1)/2 for even-number series (that's for naturals)
🏆 Exam Tips
- ✅ Three formulas: (n+1)/2, n+1, n — memorise
- ✅ Even count series → average is midway between the two middle terms
📌 Summary
- Evenly spaced → (first+last)/2
- Average = middle term
- Naturals (n+1)/2; evens n+1; odds n