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Arithmetic and Geometric Sequences Contents

1.1 Sequences 1.2 Arithmetic Sequence and Geometric Sequence 1.3 Summing a Sequence

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Arithmetic and Geometric Sequences

1.1 Sequences A sequence is a number pattern in a definite order. Each number in a sequence is called a term. For example, in the sequence of triangular numbers, 1, 3, 6, 10, 15,…, The first term is 1, which is usually denoted by T(1). Similarly, the other terms can be denoted by T(2), T(3), …etc. the (…) at the end means that the sequence continues infinitely. In the sequence of triangular numbers, the nth is given by the formula T ( n) = Content

n(n + 1) . 2

This is called the general term of the sequence

General term can be regarded as the formula generator of the sequence.

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Arithmetic and Geometric Sequences

1.1 Sequences This sequence with a common ratio between consecutive terms is called a geometric sequence ( or geometric progression, G.P.)

Apart from geometric sequence, there is another well-know sequence. This kind of sequence have a common difference between consecutive terms, and is called arithmetic sequence ( or arithmetic progression, A.P).

Content

The term ‘Geometric Progression’ and ‘Arithmetic Progression’ can be found in the past HKCEE questions on or before 1996.

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Arithmetic and Geometric Sequences

1.2 Arithmetic Sequence and Geometric Sequence A. Arithmetic Sequence Consider the sequence: 1, 4, 7, 11, 15,…. In this sequence, every term after the first term is larger than its preceding term by 3, therefore, the difference 3 is called the common difference of the sequence and this kind of sequence is called an arithmetic sequence. Let the common difference be d and the first term be a. We have: T(1) = a, T(2) = a + d, T(3) = a + 2d, … , T(n) = a + (n – 1)d, … . For an arithmetic sequence, the nth term is Content

T (n) = a + ( n − 1)d where a is the first term, d is the common difference and n is a positive integer. This is also the general term of the sequence.

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Arithmetic and Geometric Sequences

1.2 Arithmetic Sequence and Geometric Sequence B. Geometric Sequence If each term of a sequence differs from the preceding term by the same multiplying factor, this is called a geometric sequence. Moreover, the same multiplying factor is called the common ratio, usually denoted by r. If the first term of a geometric sequence is denoted by a, we have T (1) = a, T (2) = a ⋅ r , T (3) = a ⋅ r 2 , ⋅ ⋅⋅, T (n) = a ⋅ r n −1 , ⋅ ⋅ ⋅ . The nth term of a geometric sequence is T (n) = ar n −1 , Content

where a is the first term, r is the common ratio and n is a positive integer. T(n) is also called the general term of the sequence.

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Arithmetic and Geometric Sequences

1.3 Summing a Sequence The sum of the terms of a sequence is called a series. Consider the sequence: T(1), T(2), T(3), … , T(n) Let S(n) = T(1) + T(2) + T(3) + … + T(n). S(n) is called a series of n terms In other words, S(n) is the sum of the first n terms of sequence. For example, in the sequence specified by the general term T(n) = (–1)n n2, S (3) = (−1)112 + (−1) 2 2 2 + (−1)3 32 = −1 + 4 − 9 = −6 Content

S (4) = (−1)112 + (−1) 2 2 2 + (−1)3 32 + (−1) 4 4 2 = −1 + 4 − 9 + 16 = 10

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Arithmetic and Geometric Sequences

1.3 Summing a Sequence A. Summing an Arithmetic Sequence The formula for finding the sum of an arithmetic series with n terms is given by n S (n) = (a + l ). 2 When we do not know the last term l, we may substitute l = a + (n – 1)d into the formula of the sum of an arithmetic series. n S (n) = (a + a + (n – 1)d) 2 Content

Another formula for finding the sum of an arithmetic series with n terms is n S (n) = (2a + (n − 1)d ). 2

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Arithmetic and Geometric Sequences

1.3 Summing a Sequence B. Summing a Geometric Sequence For a geometric series whose first term is a and the common ratio is r, the sum to n terms is given by a (r n − 1) S (n) = where r ≠ 1 r −1

This formula works best when r > 1 so that the denominator will not be negative. If r < 1, we may use the equivalent formula: Content

a (1 − r n ) S (n) = where r ≠ 1 1− r

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Arithmetic and Geometric Sequences

1.3 Summing a Sequence B. Summing to infinity A concept which makes mathematics so fascinating is the concept of infinity. infinite sum is one of the problems explored through generations.

To find the sum to infinity S(∞) of a geometric series with –1 > r < 1, we have a simple formula S (∞) =

a . 1− r

Content “∞ “ stands for infinity.

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