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CBSE Class 11 · Mathematics

Sequences and Series

Official NCERT chapter from Mathematics (book code kemh1). ExamMaster notes are original teaching at CBSE Class 11 depth.

This lesson follows the official chapter “Sequences and Series” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 11
  • Medium level
  • 8 concepts

1Sequences

A sequence is an ordered list a1, a2, … A series is the sum a1+a2+… of those terms. 2, 6, 18 is a sequence; 2+6+18 is a series (a partial sum if it stops). Sequence is order; series is add.

Calling the list a series because it “goes on” is the heading-steal.

Figure. A sequence is an ordered list: each place n has one term. Here 2, 5, 8, 11 sit at n = 1, 2, 3, 4. Swap two terms and it is a different sequence.

How it works

  1. Write the ordered termsThe sequence.
  2. Put plus signs if a sum was askedThe series.
  3. Keep the two words apartList versus add.

2Series

A series at this class is often a named partial sum S_n. S_3 of 2+6+18 is 26. An infinite-sum claim needs the lesson’s “approaches” word if it used one; otherwise stop at S_n.

Writing ∞ as a number you reached is a miss.

Figure. A series is the sum of the terms, not the list itself. 2 + 5 + 8 + 11 = 26 is S4. The boxes are the sequence; the plus signs and the total make the series.

How it works

  1. Name how many terms you addn.
  2. Add them, or use a taught sum-formulaS_n.
  3. Refuse a finished infinity unless the lesson named a limit-sumThis class.

3Geometric Progression (G.P.)

A geometric progression multiplies by a constant r: 2, 6, 18 has r=3. a_n = a r^{n−1}. S_n = a(r^n−1)/(r−1) when r≠1, as taught. A constant add is the AP family, not this heading.

Calling 2, 6, 18 an AP with “growing d” is the family-steal.

Figure. A geometric progression multiplies by the same common ratio each step. 3, 6, 12, 24 all use r = 2. If any step used a different multiplier, the list would not be a G.P.

How it works

  1. Divide consecutive termsThe ratio.
  2. If it stays, write a_n and S_nThe GP.
  3. For 2+6+18, S_3=26; formula 2(27−1)/(3−1)=26A check.

GP 2, 6, 18

Find a_4 and S_3.

  • r3
  • a_42×27=54
  • S_32+6+18=26

Pro tip. Constant ratio; S_n has its own write when r≠1.

4Relationship Between A.M. and G.M

For positive a, b the AM is (a+b)/2 and the GM is √(ab). AM ≥ GM, with equality iff a=b (as taught). For 4 and 16, AM=10, GM=8, and 10>8. The relationship is that compare, not a new average-word.

Using (a+b)/2 as if it were √(ab) is the heading-steal.

Figure. For 4 and 16 the arithmetic mean is (4+16)/2 = 10 and the geometric mean is sqrt(4×16) = 8. On the line, AM sits to the right of GM. Equality holds only when the two numbers are equal.

How it works

  1. Write AM=(a+b)/2 and GM=√(ab)The two means.
  2. Compare AM ≥ GMThe relation.
  3. Equality only when a=b (positives as taught)The iff.

5A definition is a test you can run

Sequence / series / GP is a test: an ordered list, a sum, or a constant r. If you only say “next term”, you have a heading.

A pretty 2,6,18 banner is not the test until r is checked.

Figure. The G.P. definition is a test you run on the ratios, not a look at rising numbers. 3, 6, 12 keeps r = 2. 3, 6, 10 jumps 2 then about 1.7, so it fails.

How it works

  1. Write the list or the sum or rThe object.
  2. Name which family and the formulaThe test.
  3. Then the word has contentThe definition ran.

6Name the given before the unknown

The given is a, r (or two terms). The unknown is a_n or S_n. Copy r=3 before you use 2 as if it were r.

Using n=3 as a_n is a silent given.

Figure. Write the given a, r and n before hunting a_n. Here a = 3, r = 2, n = 4 so a4 = 3 × 2^3 = 24. The unknown is last; the three givens are first.

How it works

  1. Copy a and rThe given.
  2. Name a_n or S_nThe unknown.
  3. Then plug nGiven first.

7One worked case is enough at this class

One GP-sum is enough. A page of the same 26-clone does not add a new AM-GM.

Ten twins of r=3 are still one idea.

Figure. One finished case is enough to hold the idea at this class: 3 + 6 + 12 = 21. That is S3 for a = 3, r = 2. Do not open a general proof until this sum is clear.

How it works

  1. Find one a_n or one S_n or one AM-GM pairThe case.
  2. A second only as a checkOptional.
  3. StopThis class.

8A miss: swapping the school name for the picture

A bouncing-ball photo is a setting. The maths is a constant r. A bounce you “feel” is not r=1/2 until measured.

A spiral stair is not a GP.

Figure. A rising list is only a picture. 10, 13, 16, 19 adds 3 each step, so the school name is A.P., not G.P. Name it after the test (constant d or constant r), not after how it looks.

How it works

  1. Keep the bounce as a storyA setting.
  2. Write a and rThe maths.
  3. Then a_n or S_nDo not swap.
AM and GM of 4 and 16 are
  1. 10 and 8 — and 10>8
  2. 8 and 10
  3. Both 10

AM ≥ GM for positives; equality only if equal.

Notes

  • The official chapter title is “Sequences and Series”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • GP: a_n = a r^{n−1}
  • S_n = a(r^n−1)/(r−1) (r≠1)
  • AM=(a+b)/2, GM=√(ab), AM≥GM

Recap

Hold these pegs from the official chapter “Sequences and Series”. The wording is ExamMaster’s teaching, not a textbook recap.

Sequences
A sequence is an ordered list a1, a2, … A series is the sum a1+a2+… of those terms.
Series
A series at this class is often a named partial sum S_n.
Geometric Progression (G.P.)
A geometric progression multiplies by a constant r: 2, 6, 18 has r=3.
Relationship Between A.M. and G.M
For positive a, b the AM is (a+b)/2 and the GM is √(ab).
A definition is a test you can run
Sequence / series / GP is a test: an ordered list, a sum, or a constant r.
Name the given before the unknown
The given is a, r (or two terms). The unknown is a_n or S_n. Copy r=3 before you use 2 as if it were r.

Practise Sequences and Series

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