The 1st, 2nd and 3rd terms of a geometric progression are the 1st, 9th and 21st terms respectively of an arithmetic progression. The 1st term of each progression is 8 and the common ratio of the geometric progression is r, where r ≠ 1. Find

Let f(x) = (2 + 11x - 10x²) / ((1 + 2x)(1-2x)(2 + x)).

Find the first three terms in the expansion, in ascending powers of x, of (1 + x)5.

The first term of a progression is cos θ, where 0 < θ < ½π.

The positive numbers p and q are such that ln(p/q) = a and ln(q²p) = b. Express ln(p⁷q) in terms of a and b.

The equation of a curve is 2y² + 3xy + x = x².

Express `3sinx + 2√2cos(x + (1/4)π)` in the form `Rsin(x + α)`, where `R > 0` and `0 < α < (1/2)π`. State the exact v…