MathematicsChapter 7: Mathematical Induction and Binomial Theorem
General Term and Middle Term
T_{r+1} = nCr a^{n-r} b^r .
Learning objectives
- 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.
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). 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.
Mathematical form
n=6 even, 7 terms, one middle T4
r=3, T4 = 6C3 x3 13
n=6 even
T4 = 6C3 x
Where / symbols
- 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+1)/2 and (n+3)/2. 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 T_4 (r=3). n=5: 6 terms, middle T_3 and T_4. Write n even/odd first.
Relationship
- Tr+1 = nCr an-r br
Overview
Find r, then the middle:
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.
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.
Examples n even: one middle · n odd: two · Position from T_1
WHAT — 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 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 x^3
How to answer a general term and middle term question
- 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.
Worked example
- middle term of (x+1)^6.
- n=6 even, 7 terms, one middle T_4
- r=3, T4 = 6C3 x3 13
- = 20 x^3
In simple words
n even -> one middle. n odd -> two middles. n is the exponent, not the number of terms.
Key points
- Tr+1 = nCr an-r br
- n even: one middle; n odd: two middles
- match the exponent
- n+1 terms
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.General Term and Middle Term
Sticky Notes · Quick Revision
Core Idea
- 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.
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 T4.
- Write n even/odd first.
Worked Example
- Worked: middle term of (x+1)^6.
- n=6 even, 7 terms, one middle T_4
- r=3, T4 = 6C3 x3 13
- = 20 x^3
Common Misconception
- n even -> one middle. n odd -> two middles. n is the exponent, not the number of terms.
FBISE Exam Check
- State the rule for general term and middle term before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- 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.
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