(a)(i) Write down the expansion of \((1 + x)^{7}\) in ascending powers of x. (ii) If the coefficients of the fifth, sixth and seventh terms in the expansion...

(a)(i) Write down the expansion of \((1 + x)^{7}\) in ascending powers of x.

(ii) If the coefficients of the fifth, sixth and seventh terms in the expansion in (a)(i) above 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, 

\(\int_{1}^{5} \sqrt{(2x + 8x^{2})} \mathrm {d} x\).