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Statistics · Class 11 Commerce

Ch 9Geometric Progression — Class 11 Statistics, concept-first.

The Gujarat Board (GSHSEB) Std 11 Commerce Statistics syllabus builds on ordinary numbers arranged in a definite pattern, called a sequence. When the terms of a sequence follow a rule connecting each term to the next, the sequence is called a progression.

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Key concepts

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Geometric Progression (GP)

Imagine you drop a ball from a height of 1 metre. It bounces back to half its previous height each time. The heights of successive bounces are:

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

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Meaning of Sequences and Geometric Progression

The Gujarat Board (GSHSEB) Std 11 Commerce Statistics syllabus builds on ordinary numbers arranged in a definite pattern, called a sequence.

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General (nth) Term of a Geometric Progression

Let the first term of a GP be and the common ratio be . Then the terms are:

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Geometric Mean and Properties of a GP

Three numbers are in GP if they can be written as — writing them this way is often more convenient than because the middle term stays simply .

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Sum of n Terms of a Geometric Progression

Let denote the sum of the first terms of a GP with first term and common ratio :

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Sum to Infinity of a Geometric Progression

When a GP goes on forever () and the common ratio satisfies (i.e. ), the successive terms shrink toward zero and the running sum approaches a fixed, finite value called the sum to infinity, :

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Applications of GP in Business: Compound Interest and Annuities

Under compound interest, the amount at the end of each year is obtained by multiplying the previous year's amount by a fixed factor , where is the rate of interest per annum (as a decimal).

Exercises

+Show 6 questions6 questions
  1. Q9Find the 6th term of a GP whose first term is 7 and common ratio is 2.Free
  2. Q10The first term of a GP is 4 and its 4th term is 32. Find the common ratio.Free
  3. Q11Find the sum of the first 6 terms of the GP $2, 6, 18, \ldots$Preview
  4. Q12Find the sum to infinity of the GP whose first term is 27 and common ratio is $\tfrac13$.Preview
  5. Q13Which of the following sequences is a Geometric Progression? (a) $2, 4, 6, 8$ (b) $3, 9, 27, 81$ (c) $5, 10, 15, 20$ (d) $1, 3, 5, 7$Preview
  6. Q14A machine presently worth ₹50,000 depreciates at 10% per annum. Using the GP concept, find its value after 3 years.Preview

More questions

+Show 8 questions8 questions
  1. Example 1Find the 10th term of the GP $3, 6, 12, 24, \ldots$Free
  2. Example 2Find the common ratio and the 7th term of the GP $2, -6, 18, -54, \ldots$Free
  3. Example 3Find the sum of the first 8 terms of the GP $5, 15, 45, \ldots$Free
  4. Example 4Find the sum to infinity of the GP $8, 4, 2, 1, \ldots$Preview
  5. Example 5Express the recurring decimal $0.\overline{7}$ as a fraction using the sum to infinity of a GP.Preview
  6. Example 6Insert a Geometric Mean between 4 and 16, and verify that the resulting three numbers form a GP.Preview
  7. Example 7A sum of ₹10,000 is invested at 10% p.a. compound interest. Using the GP concept, find the amount at the end of 3 years.Preview
  8. Example 8Find the future value of an ordinary annuity of ₹1,000 per year for 4 years at 10% p.a. compounded annually, using the GP sum formula.Preview