Theory question 110
Binomial Theorem(a)(i) Write down the expansion of (1 + x)7 in escending powers of x. (ii) If the coefficients of the fifth, sixth and seventh terms in the expansion in 10(a)(i) form a linear sequence (A.P.), find the common difference of the A.P.
(b) Using the trapezium rule with ordinates at 1, 2, 3, 4 and 5, calculate, correct to two decimal places, ∫15 √(2x + 8x2) dx.
Show the model answer
Model answer
(a)(i) 1 + 7x + 21x² + 35x³ + 35x⁴ + 21x⁵ + 7x⁶ + x⁷.
(ii) Fifth, sixth, seventh coefficients 35, 21, 7: d = −14.
(b) h = 1; y = 3.1623, 6.0000, 8.8318, 11.6619, 14.4914.
Area ≈ ½[3.1623 + 14.4914 + 2(6 + 8.8318 + 11.6619)] ≈ 35.32.
Explanation
Trapezium rule: (h/2)[first + last + 2(middle ordinates)].