1.6 Sequences and Series

• Number sequences and the nth term
• Arithmetic sequences: common difference, nth term, and the sum of an arithmetic series
• Geometric sequences: common ratio, nth term, the sum of a finite geometric series and the sum to infinity
• Applications of arithmetic and geometric progressions

• determine an nth term in a number sequence
• determine terms and common difference in an arithmetic sequence and series
• compute the terms and common ratio in a geometric sequence and series
• determine the sum of terms in arithmetic and geometric series
• apply sequences and series to real-life situations
• appreciate the use of sequences and series in real life

“How is sequences and series used in real-life situations?”

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1. A rule for the nth term
   40 min · 1 practice tasks · quiz
   Use a stated rule to find a later term, and see that a short list of terms can fit more than one continuation.
2. Arithmetic sequences
   40 min · 1 practice tasks · quiz
   Recognise a common difference, including a negative difference and a zero difference, and continue the sequence.
3. The nth term of an arithmetic sequence
   40 min · 1 practice tasks · quiz
   Derive the nth term a + (n - 1)d and use it to find a later term or the position of a given term.
4. Two terms determine the sequence
   40 min · 1 practice tasks · quiz
   Use two positioned terms to build a pair of equations, then find a and d.
5. The sum of an arithmetic series
   40 min · 1 practice tasks · quiz
   Derive the sum of the first n terms by pairing the first with the last, and use it on a rising list, a falling list and the first 20 counting numbers.
6. Geometric sequences
   40 min · 1 practice tasks · quiz
   Recognise a common ratio, including a fraction and a negative ratio, and separate a geometric list from an arithmetic one.
7. The nth term of a geometric sequence
   40 min · 1 practice tasks · quiz
   Derive the nth term a times r to the power n - 1, and use it for a whole-number ratio, a fraction and a negative ratio.
8. The sum of a finite geometric series
   40 min · 1 practice tasks · quiz
   Derive the sum of the first n terms for a ratio that is not 1, and use the separate rule when every term is the same.
9. Sum to infinity
   40 min · 1 practice tasks · quiz
   Add an infinite geometric series only when the size of the ratio is less than 1, and refuse the formula when the terms do not die away.
10. Savings, seats and a light interval
    40 min · 1 practice tasks · quiz
    Use an arithmetic sum for weekly savings and theatre seats, a geometric sum to infinity for a halving prize fund, and an arithmetic term for a model light interval.
11. A fixed amount or a fixed percent
    40 min · 1 practice tasks · quiz
    Decide whether a list is arithmetic or geometric, then compare a salary that rises by 2000 shillings with one that rises by 10 percent.