Master Unit 7 Mathematical Induction & Binomial Theorem quickly and score 90+ marks in FBISE exams.
hsa notes.com notes features:
- Complete solved Exercise 7.1, 7.2, 7.3, 7.4
- Step-by-step induction proofs
- Series summation formulas
- Partial fractions & telescoping tricks
- Binomial expansions & approximations
- General term & coefficients
- Exam-focused problems & MCQs
- Free PDF download
Target: FBISE students + top schools: Aitchison College, Beaconhouse, LGS, Roots Ivy, KIPS, Garrison, OPF, IMCB, F.G. Colleges.
Key Solved Questions (Exercise 7.1):
- 1 + 2 + … + n = n(n+1)/2
- 1² + 2² + … + n² = n(n+1)(2n+1)/6
- 1³ + 2³ + … + n³ = [n(n+1)/2]²
- ∑ 1/((3k-2)(3k+1)) = n/(3n+1)
- 1² + 3² + … + (2n-1)² = n(4n²-1)/3
- 4³ + 4⁴ + … + 4ⁿ = (64/3)(4^{n-2}-1)
- ∑ 1/(k(k+1)) = n/(n+1)
- ∑ (k+2)/(k(k+1)2^k) = 1 – 1/((n+1)2^n)
- ∑ (k+4)/(k(k+1)(k+2)) = n(3n+7)/(2(n+1)(n+2))
- 7 + 77 + … + (n sevens) = 7/81 (10^{n+1} -9n -10)
- 1³ + 3³ + … + (2n+1)³ = (n+1)²(2n²+4n+1)
Key Solved Questions (Exercise 7.2):
- (2√x + 1/√x)^5 = 32x^{5/2} + 80x^{3/2} + 80x^{1/2} + 40x^{-1/2} + 10x^{-3/2} + x^{-5/2}
- (3 + y/3)^9 expansion up to y^3
- General term in (x – 1/x)^10: C(10,k) x^{10-k} (-1/x)^k
- Middle term in (a – 3/x^2)^12: C(12,6) a^6 (-3/x^2)^6
- Term independent of x in (x^2 + 1/x)^9: k=3, C(9,3)
- Ratio of coefficients: three consecutive terms ratio 6:33:110 → n=15, p/q=11/4
Key Solved Questions (Exercise 7.3):
- (1 + x/3)^{1/3} ≈ 1 + x/9 – x^2/81 + 5x^3/2187, |x|<3
- (1 – x/2)^{-1/2} ≈ 1 + x/4 + 3x^2/32 + 5x^3/128
- Approx (1.03)^{1/3} ≈ 1.0099
- Coef of x in (1 + 3x/2)^{1/3} + (1 – x/4)^{-1/2} = -95/24
- (2 + x)^{1/2} (3 – x)^{-1/3} up to x^2: constant √2 * 3^{-1/3}, etc.
- Series for y = ∑ 3/(2^{2k} k!)
Key Solved Questions (Exercise 7.4):
- Unit digit 27^{304}: 3 (cycle 4)
- 9^{205} mod 31: 9^{30}≡1, rem 4
- Frac part √26 + √10: between 6-7
- Last two digits 15^8: 25
- 10^{250} + 9^{850} – 10^{250} + 9^{850}: even
Review Exercise 7 (MCQs & Keys):
(i) Induction for all n
(iv) Middle term (a+b)^{20}: T_{11}
(vii) p in (1+x)^p, T3 coef -15/2: p=-5
Q1. Prove ∑ k/(k+1)! = 1 – 1/(n+1)!
Q3. Consecutive terms ratio 6:33:110, find n=15
Q5. Approx (3β/(5α-2β))^{-1/3} ≈ α/β
View full PDF for complete solutions, diagrams & tables.
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