1 1a i the series sum_x1100left2 frac3 x4right is an arithmetic progre

Question

the first twenty terms and the second set of twenty terms. is an arithmetic progression. Find the sum of \sum_{x=1}^{100}\left(2-\frac{3 x}{4}\right) 1. 1.(a) (i) The series (ii) Use the sum to infinity of a geometric progression to express the recurring decimal 4.16666666... as a fraction in its lowest terms. (b) (i) Use the binomial theorem to write down the first four terms as a series of ascending powers of z in the expansion of (4x + y)^8. (ii) In the binomial expansion of (3+x)^n the coefficient of x^6 equals the coefficient of x^7, Find the value of n. (c) Use the Factor Theorem to show that (r + 1) is a factor of the polynomial P(x) = 2x^4 – 5x^3 - 14x^2 + 23 x + 30. Perform the full long division to find the quotient. Hence (or otherwise) factorise P(z) into linear factors.