This page covers the following topics:
1. Arithmetic series
2. Geometric series
3. Convergent geometric sequences
4. Modelling sequences
A series is the sum of the terms of a sequence up to a certain number of terms and it is denoted by Sₙ. A series for an arithmetic sequence is given by Sₙ = n(a₁ + aₙ)/2, where n is the position of the last term, a₁ is the first term and aₙ is the last term, which can be rewritten as below.
The formula for an arithmetic series can be derived as follows. Let Sₙ = a + a + d + a + 2d + ... + a + (n − 1)d. Rewriting this in the opposite direction, gives Sₙ = a + (n − 1)d + a + (n − 2)d + ... + a. Adding these two expressions together gives 2Sₙ = 2a + (n − 1)d + 2a + (n − 1)d + ... + 2a + (n - 1)d = n(2a + (n − 1)d). Dividing by two gives the required equation.
A geometric sequence can be represented as a geometric series by taking the sum of its terms. The formula for a geometric series is given below.
The formula for a geometric series can be derived as follows. Let S = a + ar + ar² + ar³ + ... + arⁿ. Multiplying by r gives S = ar + ar² + ar³ + ... + arⁿ⁺¹. Subtracting the two expressions gives s − rs = a − arⁿ⁺¹. Factorising gives s(1 − r) = a(1 − rⁿ⁺¹). Rearranging gives the required result.
An infinite series has an infinite number of terms. The partial sum (Sₙ) of these terms is the sum of the first n terms. If Sₙ approaches a finite limit, known as the sum to infinity, the series is convergent. If it doesn't, it is said to be a divergent series. In a geometric series, if |r| < 1, the sum to infinity can be calculated with the following formula.
Sequences and series can be used to model real-life scenarios. When the model involves a change by a fixed amount, arithmetic sequences and series can be used. When the model involves a change by a factor or a percentage, geometric sequences and series can be used.
Lucy is learning to play the guitar. She notices she leaves every lesson with two more printed off music sheets than she did the last week. Given Lucy ended her first lesson with one music sheet as homework, how many sheets will she have in total by her fourth lesson?
aₙ = 2n − 1
a₁ = 2(1) − 1 = 1
a₄ = 2(4) − 1 = 7
S₄ = 4(1 + 7)/2 = 16
The number of assignments Jake has in a week follows an arithmetic sequence, where the number of assignments increases by 1 every week. Given that in the first week of university, he has 2 assignments, calculate the total number of assignments he will have in the first 4 weeks of university.
S₄ = (4/2)(2 × 2 + (4 − 1)(1)) = 14
Explain why the sum to infinity of the following geometric series cannot be calculated: 3 + 9 + 27 + ... .
The common ratio is given by 9/3 = 3.
Since r = 3 > 1, the geometric series is divergent and therefore the sum to infinity cannot be calculated.
What is the sum of the first ten terms of the geometric sequence 8, −24, 72, −216, ...?
The common ratio is given by −24/8 = −3.
Using the formula for the sum of a geometric sequence, sum = 8 × (1 − (−3)¹⁰)/(1 − (−3)) = −118096.
Given that the sum of the first n terms of the following sequence is 65535, calculate the value of n: 3, 12, 48, 192, ...
The common ratio is given by 12/3 = 4.
Using the formula for the sum of a geometric series, 65535 = 3((1 − 4^n)/(1 − 4)
So, 4^n = 65536 and taking logs of both sides gives n = log(65536)/log(4) = 8.
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