Coaching Centre
Chapter 1: Sequences and series Notes
Arithmetic and geometric sequences, their nth terms, partial sums and sigma notation, the sum of an AP and of a GP, limiting sums, and recurring decimals.
Sections
-
1A Sequences and how to specify them Mastered
Terms of a sequence, finite and infinite sequences, specifying a sequence by listing, by a formula for the nth term, or by a recursive rule, and solving T(n) = k for n.
-
1B Arithmetic sequences Mastered
Arithmetic sequences (APs): the common difference, testing for an AP, the formula T(n) = a + (n − 1)d, counting the terms of an AP, and the link with linear functions.
-
1C Geometric sequences Mastered
Geometric sequences (GPs): the common ratio, testing for a GP, the formula T(n) = ar^(n−1), negative ratios and alternating signs, and finding which term reaches a given size.
-
1D Solving problems involving APs and GPs Mastered
Three terms in arithmetic or geometric progression, the arithmetic and geometric means, simultaneous equations from two given terms, and using logarithms to find when a GP passes a value.
-
1E Adding up the terms of a sequence Mastered
Series and partial sums S(n), recovering the terms from the partial sums with T(n) = S(n) − S(n−1), and reading and writing sums in sigma notation.
-
1F Summing an arithmetic series Mastered
The two formulae for the sum of an arithmetic series, S(n) = n/2 (a + ℓ) and S(n) = n/2 (2a + (n − 1)d), and solving problems including finding how many terms give a given sum.
-
1G Summing a geometric series Mastered
The formula for the sum of a geometric series in its two forms, sums with negative and fractional ratios, sums in sigma notation, and using logarithms to find how many terms exceed a given sum.
-
1H The limiting sum of a geometric series Mastered
When an infinite geometric series has a limiting sum, the condition −1 < r < 1, the formula S = a/(1 − r), and problems including finding the values of x for which a limiting sum exists.
-
1I Recurring decimals and geometric series Mastered
Writing a recurring decimal as an infinite geometric series and using the limiting sum to convert it to a fraction, including decimals where the repeating block starts later.