Approximating with an expansion, progressions, and coefficients in progressions

Last time

Before we start: last time, in The binomial expansion, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Patterns in numbers

Two ideas from series, joined in Paper 1 questions. The binomial expansion gives the terms of a bracket raised to a power, and a few terms already give an excellent approximation. Arithmetic and geometric progressions add by a steady gap or multiply by a steady ratio, and a question may even make the coefficients of an expansion form a progression.

The binomial expansion

The expansion of a plus b to the n has terms n choose r, times a to the n minus r, times b to the r. The coefficient comes first. For 2 minus three halves x to the power 5, the first three terms are 32, then 5 times 16 times minus three halves x, then 10 times 8 times nine quarters x squared. Keep the signs inside the bracket with the term. Simplify each coefficient fully.

An approximation

To find 1.985 to the 5th, write 1.985 as 2 minus three halves times 0.01. So x is 0.01. Substitute into the first three terms: 32 minus 1.2 plus 0.018, which is 30.818. The next term is tiny. The approximation is close to the true value.