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Sum of n Terms of Geometric Progression (G.P.)

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Introduction

Geometric Progression (G.P.) में हर term पिछले term को एक fixed ratio r से multiply करके मिलता है। कई questions में G.P. के पहले n terms का total (Sₙ) निकालना होता है - खासकर finance, population growth, और interest-based problems में। यह formula पूरे series का total जल्दी और बिना हर term को लिखे निकालने में मदद करता है।

Pattern: Sum of n Terms of Geometric Progression (G.P.)

Pattern: Sum of n Terms of Geometric Progression (G.P.)

G.P. के पहले n terms का sum (Sₙ) इस प्रकार होता है:

जब r ≠ 1: Sₙ = a × (rⁿ - 1) / (r - 1)

जब r < 1 (decreasing G.P.): Sₙ = a × (1 - rⁿ) / (1 - r)

यहाँ a = first term, r = common ratio, और n = number of terms।

Step-by-Step Example

Question

G.P. 3, 6, 12, 24, 48 के पहले 5 terms का sum निकालें।

Solution

  1. Step 1: a, r और n पहचानें

    First term a = 3
    Common ratio r = 6 ÷ 3 = 2
    Number of terms n = 5

  2. Step 2: r ≠ 1 के लिए formula apply करें

    Sₙ = a × (rⁿ - 1) / (r - 1)

    Substitute: S₅ = 3 × (2⁵ - 1) / (2 - 1)

  3. Step 3: Simplify

    S₅ = 3 × (32 - 1) / 1 = 3 × 31 = 93

  4. Final Answer:

    पहले 5 terms का sum = 93

  5. Quick Check:

    3 + 6 + 12 + 24 + 48 = 93 ✅

Quick Variations

1. अगर series decreasing हो (r < 1) → formula बदलकर Sₙ = a(1 - rⁿ)/(1 - r)।

2. जब r = 1 हो → हर term same होगा → Sₙ = n × a।

3. Sum और ratio दिए हों तो इसी formula को rearrange करके n भी निकाला जा सकता है।

Trick to Always Use

  • Step 1: पहले r की value चेक करें (1 से बड़ा या छोटा)।
  • Step 2: r > 1 हो तो formula (rⁿ - 1)/(r - 1) और r < 1 हो तो (1 - rⁿ)/(1 - r) use करें।
  • Step 3: Powers (rⁿ) पहले calculate कर लें - इससे steps आसान होते हैं।

Summary

G.P. में:

  • Formula: Sₙ = a × (rⁿ - 1)/(r - 1) (जब r > 1).
  • Alternate: Sₙ = a × (1 - rⁿ)/(1 - r) (जब r < 1).
  • Special case → r = 1 हो तो Sₙ = n × a।
  • Exponent calculation ध्यान से करें - values बहुत बड़ी भी हो सकती हैं।

Practice

(1/5)
1. Find the sum of the first 4 terms of the G.P.: 2, 4, 8, 16.
easy
A. 32
B. 28
C. 34
D. 30

Solution

  1. Step 1: Identify a, r and n

    First term a = 2, common ratio r = 4 ÷ 2 = 2, number of terms n = 4.

  2. Step 2: Apply the sum formula for r ≠ 1

    Sₙ = a × (rⁿ - 1)/(r - 1). Substitute: S₄ = 2 × (2⁴ - 1)/(2 - 1).

  3. Step 3: Simplify

    S₄ = 2 × (16 - 1)/1 = 2 × 15 = 30.

  4. Final Answer:

    The sum of the first 4 terms is 30 → Option D.

  5. Quick Check:

    Direct addition: 2 + 4 + 8 + 16 = 30 ✅

Hint: When r = 2, total = a × (2ⁿ - 1).
Common Mistakes: Forgetting to subtract 1 from rⁿ before multiplying by a.
2. Find the sum of the first 5 terms of the G.P.: 3, 6, 12, 24, 48.
easy
A. 90
B. 93
C. 96
D. 99

Solution

  1. Step 1: Identify a, r, and n

    First term a = 3, ratio r = 2, number of terms n = 5.

  2. Step 2: Apply the formula Sₙ = a × (rⁿ - 1)/(r - 1)

    S₅ = 3 × (2⁵ - 1)/(2 - 1) = 3 × (32 - 1) = 3 × 31 = 93.

  3. Final Answer:

    The sum of the first 5 terms is 93 → Option B.

  4. Quick Check:

    3 + 6 + 12 + 24 + 48 = 93 ✅

Hint: Use a × (2ⁿ - 1) when r = 2 to speed up calculation.
Common Mistakes: Dropping parentheses around (rⁿ - 1).
3. Find the sum of the first 3 terms of the G.P.: 81, 27, 9.
easy
A. 119
B. 121
C. 117
D. 123

Solution

  1. Step 1: Identify a, r and n

    First term a = 81, ratio r = 27 ÷ 81 = 1/3, number of terms n = 3.

  2. Step 2: Use the decreasing-G.P. formula

    For r < 1 use Sₙ = a × (1 - rⁿ)/(1 - r). Substitute: S₃ = 81 × (1 - (1/3)³)/(1 - 1/3).

  3. Step 3: Simplify step-by-step

    r³ = 1/27 → 1 - r³ = 26/27. Denominator 1 - r = 2/3. So S₃ = 81 × (26/27) × (3/2) = 117.

  4. Final Answer:

    The sum of the first 3 terms is 117 → Option C.

  5. Quick Check:

    Direct addition: 81 + 27 + 9 = 117 ✅

Hint: For fractional ratios, compute (1 - rⁿ)/(1 - r) to avoid sign errors.
Common Mistakes: Applying (rⁿ - 1) instead of (1 - rⁿ) when r < 1.
4. If the first term of a G.P. is 5 and the common ratio is 3, find the sum of the first 6 terms.
medium
A. 1820
B. 1825
C. 1828
D. 1830

Solution

  1. Step 1: Record given values

    First term a = 5, ratio r = 3, number of terms n = 6.

  2. Step 2: Apply Sₙ = a × (rⁿ - 1)/(r - 1)

    S₆ = 5 × (3⁶ - 1)/(3 - 1) = 5 × (729 - 1)/2 = 5 × 728/2 = 5 × 364 = 1820.

  3. Final Answer:

    The sum of the first 6 terms is 1820 → Option A.

  4. Quick Check:

    Compute r⁶ - 1 = 728 and divide by 2, then multiply by 5 → 1820 ✅

Hint: Compute rⁿ first, then use the (rⁿ - 1)/(r - 1) factor to avoid intermediate rounding.
Common Mistakes: Forgetting to divide by (r - 1) after computing rⁿ - 1.
5. Find the sum of the first 8 terms of the G.P.: 256, 128, 64, …
medium
A. 510
B. 510.5
C. 511
D. 512

Solution

  1. Step 1: Identify a, r and n

    First term a = 256, ratio r = 128 ÷ 256 = 1/2, number of terms n = 8.

  2. Step 2: Use Sₙ = a × (1 - rⁿ)/(1 - r) for r < 1

    S₈ = 256 × (1 - (1/2)⁸)/(1 - 1/2).

  3. Step 3: Simplify carefully

    (1/2)⁸ = 1/256 → 1 - rⁿ = 255/256. Denominator 1 - r = 1/2. So S₈ = 256 × (255/256) × 2 = 255 × 2 = 510.

  4. Final Answer:

    The sum of the first 8 terms is 510 → Option A.

  5. Quick Check:

    The infinite-limit sum would be 256/(1 - 1/2) = 512; for 8 terms we get 512 - 2 = 510 (close to the limit) ✅

Hint: For r = 1/2, Sₙ = a × (1 - 1/2ⁿ)/(1/2) = 2a × (1 - 1/2ⁿ).
Common Mistakes: Using the r > 1 formula instead of the r < 1 variant.