Sequences and series

(a) A sequence is defined by u₁ = 2, u_n+1 = 3u_n - k, where k is a constant. Given that u₃ = 8, find the value of k. (b) Find the value of Σ_r=1¹⁰ (2^r + 3).

(a) Find the binomial expansion of √(4 - x), in ascending powers of x, up to and including the term in x². Give each coefficient as a simplified fraction. (b) State the range of values of x for which the expansion is valid. (c) Use your expansion with x = 0.04 to find an estimate for √3.96, giving your answer to 6 decimal places.

Jo saves money each month. She saves £50 in the first month and increases the amount by £5 each month after that. (a) Find the total amount she has saved after 24 months. (b) Find the month in which her total savings first exceed £5000.