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The assignment carries a total number of 100 marks. The marks for each question or part are as indicated.

(1) (a) [5 marks] The first term of an arithmetic sequence is 7, its last term in 70 and its sum is 385. Find the number of terms in the sequence and the common difference

(b) A geometric series is given as follows:

(i) [5 marks] Find the set of values of x such that the sum to infinity of the series exists (in other words, the series converges).

(ii) [5 marks] Find the value of x for which the sum to infinity
of the series S∞ is 43.

(c) [5 marks] An arithmetic sequence {an} ∞ n=1 of positive terms is such that twice of the sum of the first nine terms is equal to the sum of the next nine terms. Furthermore, a1, 20, a16 forms a geometric sequence. Find the first term and the common difference of the arithmetic sequence.

(2) The sum Sn of the first n terms of a sequence {ui}
∞ i=1 is given by Sn = n(2n + 2024).
(i) [5 marks] Find un in terms of n.
(ii) [5 marks] Find un+1 in terms of un.

(3) The sequence {an} is defined as follows:
a1 = 2, an+1 = 2(a1 + a2 + ⋅ ⋅ ⋅ + an) for n = 1, 2, 3 . . .
(i) [5 marks] Evaluate the numerical value of a2, a3 and a4.
(ii) [5 marks] Find the ratio of an+1
an where n ≥ 2. Is {an}∞n=2ageometric sequence?
(iii) [5 marks] Find the sum of the n terms a1 + a2 + ⋅ ⋅ ⋅ + an.

(5) (a) [5 marks] Let n be a positive integer with n > 1. Suppose that the coefficient of x 3 in the expansion of (1 + x) 2n and the coefficient of x 2 in the expansion of (1 + 10x) n are equal. Find the value of n.

(b) [5 marks] Let k > 0 be a constant. Suppose the coefficient of x2 in the expansion of

(6) Consider the binomial expansion (1 − x)−2

(i) [5 marks] Write down its first three terms in ascending powersof x.

(ii) [5 marks] Find the coefficient of xn.

(iii) [5 marks] Find the range of values of x for which the expansion is valid.

(iv) [5 marks] Hence, or otherwise, find the value of

∞∑n=1n2

(7) (i) [5 marks] Expand (1−x1+x)n in ascending powers of x up to and including the term in x2.

(ii) [5 marks] State the set of values of x for which the series expansion is valid.
(iii) [5 marks] Hence find an approximation to the fourth root of 19 21 , in the form p q, where p and q are positive integers with no common factors.

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