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Series-Based Approximation

Introduction

Series-based approximation problems आपकी ability को test करते हैं कि आप arithmetic (AP), geometric (GP) या special series जैसी sequences को कितनी जल्दी simplify कर सकते हैं। हर term को एक-एक करके calculate करने की बजाय, आप formulas या shortcuts का उपयोग करके series को evaluate या approximate करते हैं।

Pattern: Series-Based Approximation

Pattern

मुख्य idea: Standard series formulas (AP, GP, sum of squares, sum of cubes) या simple patterns पहचानकर totals को जल्दी compute करें।

  • AP sum formula: Sn = n/2 × (first term + last term)
  • पहले n natural numbers का sum: n(n+1)/2
  • पहले n odd numbers का sum: n²
  • पहले n even numbers का sum: n(n+1)
  • GP sum formula: Sn = a(1-rn)/(1-r)

Step-by-Step Example

Question

पहले 5 odd numbers का sum निकालें: 1 + 3 + 5 + 7 + 9.

Options:

  • A) 25
  • B) 23
  • C) 27
  • D) 21

Solution

  1. Step 1: Formula याद करें

    पहले n odd numbers का sum = n².

  2. Step 2: n = 5 के लिए compute करें

    Sum = 5² = 25.

  3. Final Answer:

    25 → Option A.

  4. Quick Check:

    Direct addition → 1 + 3 + 5 + 7 + 9 = 25 ✅

Quick Variations

1. पहले n even numbers का sum = n(n+1).

2. Arithmetic series जैसे 2 + 4 + 6 + … + 20.

3. Geometric series approximations जब ratio < 1 हो।

4. बहुत बड़े n के लिए approximation (जैसे पहले 1000 natural numbers का sum)।

Trick to Always Use

  • Step 1 → Series का type पहचानें (AP, GP, odd, even, squares, cubes)।
  • Step 2 → Term-by-term जोड़ने की बजाय सही standard formula apply करें।
  • Step 3 → जब n बहुत बड़ा हो तो approximation का उपयोग करें।

Summary

Summary

  • Pattern को जल्दी पहचानें (odd, even, AP, GP)।
  • Manual addition की जगह shortcut formulas use करें।
  • Exact values जरूरी न हों तो approximation helpful है।
  • Formula verify करने के लिए हमेशा कोई छोटा example देखें।

याद रखने लायक example:
AP formula Sn = n/2 × (first term + last term) का उपयोग करें; जैसे पहले 100 natural numbers का sum = 100×101/2 = 5050.

Practice

(1/5)
1. Find the sum of the first 6 odd numbers.
easy
A. 36
B. 30
C. 28
D. 32

Solution

  1. Step 1: Recall the formula

    Sum of first n odd numbers = n².

  2. Step 2: Compute for n = 6

    6² = 36.

  3. Final Answer:

    36 → Option A.

  4. Quick Check:

    1 + 3 + 5 + 7 + 9 + 11 = 36 ✅
Hint: n odd numbers always sum to n².
Common Mistakes: Adding terms manually and making mistakes.
2. Find the sum of the first 5 even numbers.
easy
A. 30
B. 25
C. 28
D. 32

Solution

  1. Step 1: Recall the formula

    Sum of first n even numbers = n(n+1).

  2. Step 2: Compute for n = 5

    5 × 6 = 30.

  3. Final Answer:

    30 → Option A.

  4. Quick Check:

    2 + 4 + 6 + 8 + 10 = 30 ✅
Hint: Sum of n even numbers = n(n+1).
Common Mistakes: Using formula for odd numbers instead of even.
3. Find the sum of the first 10 natural numbers.
easy
A. 50
B. 55
C. 60
D. 45

Solution

  1. Step 1: Recall the formula

    Sum of first n natural numbers = n(n+1)/2.

  2. Step 2: Compute for n = 10

    (10 × 11) ÷ 2 = 55.

  3. Final Answer:

    55 → Option B.

  4. Quick Check:

    1 + 2 + … + 10 = 55 ✅
Hint: Use n(n+1)/2 for natural numbers.
Common Mistakes: Forgetting to divide by 2.
4. Find the sum of the first 20 natural numbers.
medium
A. 200
B. 220
C. 230
D. 210

Solution

  1. Step 1: Recall the formula

    Sum of first n natural numbers = n(n+1)/2.

  2. Step 2: Compute for n = 20

    (20 × 21) ÷ 2 = 420 ÷ 2 = 210.

  3. Final Answer:

    210 → Option D.

  4. Quick Check:

    Pairing method: (1+20)=21, (2+19)=21, … 10 pairs × 21 = 210 ✅
Hint: n(n+1)/2 gives the exact sum; pairing terms gives a quick check.
Common Mistakes: Forgetting to divide by 2 or miscounting pairs.
5. Find the sum of the arithmetic series: 5 + 10 + 15 + … + 50.
medium
A. 250
B. 260
C. 275
D. 280

Solution

  1. Step 1: Identify the series

    First term a = 5, last term l = 50, common difference d = 5.

  2. Step 2: Count terms

    n = (l - a)/d + 1 = (50 - 5)/5 + 1 = 10.

  3. Step 3: Apply AP sum formula

    S = n/2 × (a + l) = 10/2 × (5 + 50) = 5 × 55 = 275.

  4. Final Answer:

    275 → Option C.

  5. Quick Check:

    Average term = (5 + 50)/2 = 27.5; 27.5 × 10 = 275 ✅
Hint: AP sum = average term × number of terms.
Common Mistakes: Forgetting to add +1 when counting terms.

Mock Test

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