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General Term and Middle Term
T_{r+1} = nCr a^{n-r} b^r . To find a particular power, set the exponent of x (or a) equal to the wanted power and solve for r (r must be an integer 0 to n). Middle term: if n+1 is odd, n even, there is one middle term, position (n/2)+1. If n odd, two middle terms, positions n+12 and n+32. Meaning: count terms from 1 to n+1, pick the centre.
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General Term and Middle Term
What is it?
T_{r+1} = nCr a^{n-r} b^r . To find a particular power, set the exponent of x (or a) equal to the wanted power and solve for r (r must be an integer 0 to n). Middle term: if n+1 is odd, n even, there is one middle term, position (n/2)+1. If n odd, two middle terms, positions n+12 and n+32. Meaning: count terms from 1 to n+1, pick the centre.
In simple words
- Tr+1 = nCr an-r br .
- To find a particular power, set the exponent of x (or a) equal to the wanted power and solve for r (r must be an integer 0 to n).
- Middle term: if n+1 is odd, n even, there is one middle term, position (n/2)+1.
- n even -> one middle. n odd -> two middles. n is the exponent, not the number of terms.
Important terms
GENERAL TERM AND MIDDLE TERM: T_{r+1} = nCr a^{n-r} b^r . To find a particular power, set the exponent of x (or a) equal to the wanted power and solve for r (r must be an integer 0 to n). Middle term: if n+1 is odd, n even, there is one middle term, position (n/2)+1. If n odd, two middle terms, positions n+12 and n+32. Meaning: count terms from 1 to n+1, pick the centre.
1. PARTICULAR TERM: Want x^3 in (1+x)^10: 10-r=3 wait that is if a=1,b=x then b^r = x^r so r=3, T_4 = 10C3 x^3. Watch which letter carries x.
2. MIDDLE: n=6: 7 terms, middle T4 (r=3). n=5: 6 terms, middle T3 and T_4. Write n even/odd first.
What you should learn
- State the main ideas and notation used in general term and middle term.
- Apply the rules of general term and middle term in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- Solve for r
- r integer in range
- Include the coefficient number
PARTICULAR TERM
Want x^3 in (1+x)^10: 10-r=3 wait that is if a=1,b=x then b^r = x^r so r=3, T_4 = 10C3 x^3. Watch which letter carries x.
MIDDLE
n=6: 7 terms, middle T4 (r=3). n=5: 6 terms, middle T3 and T_4. Write n even/odd first.
Recommended for Class 11
Full school-level explanation
T_{r+1} = nCr a^{n-r} b^r . To find a particular power, set the exponent of x (or a) equal to the wanted power and solve for r (r must be an integer 0 to n). Middle term: if n+1 is odd, n even, there is one middle term, position (n/2)+1. If n odd, two middle terms, positions n+12 and n+32. Meaning: count terms from 1 to n+1, pick the centre.
Find r, then the middle:
WHAT — GENERAL TERM AND MIDDLE TERM
T_{r+1} = nCr a^{n-r} b^r . To find a particular power, set the exponent of x (or a) equal to the wanted power and solve for r (r must be an integer 0 to n). Middle term: if n+1 is odd, n even, there is one middle term, position (n/2)+1. If n odd, two middle terms, positions n+12 and n+32. Meaning: count terms from 1 to n+1, pick the centre.
RULE — 1. PARTICULAR TERM
Want x^3 in (1+x)^10: 10-r=3 wait that is if a=1,b=x then b^r = x^r so r=3, T_4 = 10C3 x^3. Watch which letter carries x.
METHOD — 2. MIDDLE
n=6: 7 terms, middle T4 (r=3). n=5: 6 terms, middle T3 and T_4. Write n even/odd first.
WORKED EXAMPLE
Worked: middle term of (x+1)^6. n=6 even, 7 terms, one middle T4 r=3, T4 = 6C3 x3 13 = 20 x3
PARTICULAR TERM
Want x^3 in (1+x)^10: 10-r=3 wait that is if a=1,b=x then b^r = x^r so r=3, T_4 = 10C3 x^3. Watch which letter carries x.
Remember
- match the exponent
Examples
- Solve for r
- r integer in range
- Include the coefficient number
MIDDLE
n=6: 7 terms, middle T4 (r=3). n=5: 6 terms, middle T3 and T_4. Write n even/odd first.
Remember
- n+1 terms
Examples
- n even: one middle
- n odd: two
- Position from T_1
How to answer a general term and middle term question
How it works
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Remember
- Tr+1 = nCr an-r br
- n even: one middle; n odd: two middles
- 20 x^3
- n even/odd
- match the exponent
- n+1 terms
- Tr+1 = nCr an-r br
- n even: one middle; n odd: two middles
How the ideas connect
Worked: middle term of (x+1)^6. n=6 even, 7 terms, one middle T4 r=3, T4 = 6C3 x3 13 = 20 x3
Deeper understanding
Go beyond memorising General Term and Middle Term. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: middle term of (x+1)^6. n=6 even, 7 terms, one middle T4 r=3, T4 = 6C3 x3 13 = 20 x3
Common mistakes
Exam focus
Core Idea
1. PARTICULAR TERM
2. MIDDLE
Worked Example
Exam focus
Quick revision
- Tr+1 = nCr an-r br
- n even: one middle; n odd: two middles
- 20 x^3
- n even/odd
- GENERAL TERM AND MIDDLE TERM: Tr+1 = nCr an-r br . To find a particular power, set the exponent of x (or a) equal to the wa
- 1. PARTICULAR TERM: Want x^3 in (1+x)^10: 10-r=3 wait that is if a=1,b=x then b^r = x^r so r=3, T_4 = 10C3 x^3. Watch wh
n even -> one middle. n odd -> two middles. n is the exponent, not the number of terms.
For long answers on General Term and Middle Term: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- Tr+1 = nCr an-r br
- n even: one middle; n odd: two middles
- 20 x^3
- n even/odd
Important formulas
- Tr+1 = nCr an-r br
- n even: one middle; n odd: two middles
- 20 x^3
Common exam traps
- n even -> one middle. n odd -> two middles. n is the exponent, not the number of terms.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
n even -> one middle. n odd -> two middles. n is the exponent, not the number of terms.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
n even -> one middle. n odd -> two middles. n is the exponent, not the number of terms.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
Tr+1 = nCr an-r br
n even: one middle; n odd: two middles
20 x^3
n even/odd
Tr+1 = nCr an-r br . To find a particular power, set the exponent of x (or a) equal to the wanted power and solve for r (r must be an integer 0 to n). Mi
Tr+1 = nCr an-r br
- Definition
- Tr+1 = nCr an-r br . To find a particular power, set the exponent of x (or a) equal to the wanted power and solve for r (r must be an integer 0 to n). Mi
- Formula
- Tr+1 = nCr an-r br
- Example
- Solve for r
- Critical warning
- n even -> one middle. n odd -> two middles. n is the exponent, not the number of terms.
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Exam Practice
Define
Q1. Define General Term and Middle Term.
Explain
Q2. Explain: PARTICULAR TERM.
Example
Q3. Give one clear example of General Term and Middle Term and say why it fits.
Mistake
Q4. State one common mistake students make about General Term and Middle Term.
Exam note
Q5. Write a short exam note on General Term and Middle Term: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.