Mathematics and Statistics · Class 11 Commerce
Ch 4Sequences and Series — Class 11 Mathematics and Statistics, concept-first.
A sequence is an ordered list of numbers written according to some definite rule — for example or . Each number in the list is a term, and the terms are written (some books use ), where denotes the -th term — the term in position .
Key concepts
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Special Series — Sums of Powers of Natural Numbers
, , , with sums split by linearity: .
Most relevant Q&A
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Sequences and Series — The Basic Idea
A sequence is an ordered list of numbers written according to some definite rule — for example or . Each number in the list is a term, and the terms are written (some books use ), where denotes the -t…
Arithmetic Progression: nth Term
An Arithmetic Progression (AP) is a sequence in which each term after the first is obtained by adding a fixed number to the previous term.
Sum of n Terms of an AP
The sum of the first terms of an AP, , has a direct closed-form formula — useful whenever a running total is needed (total savings over several months, total output over several years).
Geometric Progression: nth Term
A Geometric Progression (GP) is a sequence in which each term after the first is obtained by multiplying the previous term by a fixed non-zero number.
Sum of n Terms of a GP, and Sum to Infinity
The sum of the first terms of a GP also has a closed form (for ).
Harmonic Progression (HP)
A Harmonic Progression (HP) is a sequence whose reciprocals form an Arithmetic Progression. That is, is an HP precisely when is an AP (with no term equal to zero).
Arithmetic, Geometric and Harmonic Means
For two positive numbers and , three different "averages" arise naturally — one from each progression.
Special Series: Sums of Powers of Natural Numbers
Three standard series of the natural numbers appear constantly in statistics (they reappear in means, variance and moments), and the syllabus states them by name using the summation symbol (sigma), wh…
Exercises
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- Q11Find the sum $\sum_{r=1}^{30} r = 1 + 2 + 3 + \cdots + 30$.Free
- Q12Evaluate $\sum_{r=1}^{12} r^2 = 1^2 + 2^2 + \cdots + 12^2$.Free
- Q13Evaluate $\sum_{r=1}^{5} r^3 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3$.Preview
- Q14A person saves ₹500 in the first month and increases the monthly saving by ₹100 every subsequent month. Find the total amount saved over the…Preview
- Q15The population of a town is $10{,}000$ and grows at a constant rate of $10\%$ per year. Treating the present population as the 1st term of a…Preview
More questions
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- Example 1Find the 20th term of the Arithmetic Progression $3, 7, 11, 15, \ldots$Free
- Example 2Which term of the Arithmetic Progression $5, 11, 17, 23, \ldots$ is equal to $95$?Free
- Example 3Three numbers are in Arithmetic Progression. Their sum is $15$ and their product is $80$. Find the numbers.Free
- Example 4Find the sum of the first $25$ terms of the Arithmetic Progression $2, 5, 8, 11, \ldots$Preview
- Example 5Find the 6th term of the Geometric Progression $2, 6, 18, 54, \ldots$Preview
- Example 6Find the sum of the first $5$ terms of the Geometric Progression $5, 15, 45, \ldots$Preview
- Example 7Three numbers are in Geometric Progression. Their product is $216$ and their sum is $26$. Find the numbers.Preview
- Example 8Find the sum to infinity of the Geometric Progression $8 + 4 + 2 + 1 + \cdots$Preview
- Example 9Find the 10th term of the Harmonic Progression $\frac{1}{6}, \frac{1}{9}, \frac{1}{12}, \frac{1}{15}, \ldots$Preview
- Example 10For the numbers $4$ and $9$, find the Arithmetic Mean (A), Geometric Mean (G) and Harmonic Mean (H), and verify that $G^2 = A \times H$ and…Preview