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Business Mathematics and Basic Statistics · Class 11 Commerce

Ch 7Arithmetic and Geometric Progressions — Class 11 Business Mathematics and Basic Statistics, concept-first.

A sequence is simply a list of numbers written in a definite order, following some rule — for example or . Each number in the list is called a term, and the terms are written (or sometimes ), where denotes the -th term.

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Key concepts

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Sum of the First n Terms of an AP

, derived by pairing the first and last terms (and so on inward), each pair summing to .

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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.

1

Sequences and Progressions — Introduced Through Examples

A sequence is simply a list of numbers written in a definite order, following some rule — for example or .

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Arithmetic Progression: First Term and Common Difference

An Arithmetic Progression (AP) is a sequence in which the difference between any term and the one immediately before it is always the same constant.

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Geometric Progression: First Term and Common Ratio

A Geometric Progression (GP) is a sequence in which the ratio of any term to the one immediately before it is always the same constant (and no term is zero).

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The General (nth) Term of an AP and a GP

Once a progression's first term and common difference/ratio are known, its general term (the -th term, ) can be written directly as a formula, without listing every term up to it.

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Finding a Term's Value or Its Position

The general-term formula can be used in two directions, depending on what is given and what is asked:

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Some Important Series

Beyond AP and GP, three specific series of natural numbers are important enough in business mathematics and statistics (they reappear in later work on means and moments) to be stated here by name, usi…

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Sum of the First n Terms of an AP and a GP

The sum of the first terms of a progression, denoted , also has a direct formula for both AP and GP — useful whenever a running total (total savings over several months, total production over several…

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Numerical Verification of the Three Summation Formulae

"Verifying" one of §6's three summation formulas for a given positive integer means: compute the left-hand side by direct addition, compute the right-hand side by substituting the same into the formul…

Exercises

+Show 4 questions4 questions
  1. Q10Find the sum of all multiples of 5 between 1 and 100 (inclusive).Free
  2. Q11Find the sum of the first 6 terms of the Geometric Progression $64, 32, 16, 8, \ldots$Free
  3. Q12A man saves ₹200 in the first month and increases his savings by ₹50 every subsequent month. Find his total savings over the first 12 months…Preview
  4. Q13The population of a town was 8,000 in 2020 and grows at a constant rate of 5% per year. Taking 2020 as the 1st term of a Geometric Progressi…Preview

More questions

+Show 9 questions9 questions
  1. Example 1Find the 10th term of the Arithmetic Progression $3, 7, 11, 15, \ldots$Free
  2. Example 2Which term of the Arithmetic Progression $5, 11, 17, \ldots$ is equal to $65$?Free
  3. Example 3Find the 6th term of the Geometric Progression $2, 6, 18, 54, \ldots$Free
  4. Example 4Which term of the Geometric Progression $3, 6, 12, \ldots$ is equal to $384$?Preview
  5. Example 5Find the sum of the first 20 terms of the Arithmetic Progression $4, 7, 10, \ldots$Preview
  6. Example 6Find the sum of the first 6 terms of the Geometric Progression $3, 6, 12, \ldots$Preview
  7. Example 7Verify that $\sum_{i=1}^{6} i = \dfrac{n(n+1)}{2}$ holds for $n=6$.Preview
  8. Example 8Verify that $\sum_{i=1}^{5} i^2 = \dfrac{n(n+1)(2n+1)}{6}$ holds for $n=5$.Preview
  9. Example 9Verify that $\sum_{i=1}^{4} i^3 = \left(\dfrac{n(n+1)}{2}\right)^2$ holds for $n=4$.Preview