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Given the geometric series: \( \frac{1}{5} + \frac{1}{15} + \frac{1}{45} + \ldots \) 2.1.1 Is this a convergent geometric series? Justify your answer with the necessary calculations - NSC Mathematics - Question 2 - 2023 - Paper 1
Question 2
Given the geometric series: \( \frac{1}{5} + \frac{1}{15} + \frac{1}{45} + \ldots \) 2.1.1 Is this a convergent geometric series? Justify your answer with the nece... show full transcript
Worked Solution & Example Answer:Given the geometric series: \( \frac{1}{5} + \frac{1}{15} + \frac{1}{45} + \ldots \) 2.1.1 Is this a convergent geometric series? Justify your answer with the necessary calculations - NSC Mathematics - Question 2 - 2023 - Paper 1
Step 1
2.1.1 Is this a convergent geometric series? Justify your answer with the necessary calculations.
Answer
To determine if the series is convergent, we need to find the common ratio ( r ). The first term is ( a = \frac{1}{5} ) and the second term is ( \frac{1}{15} ). The common ratio can be calculated as follows:
[ r = \frac{\text{second term}}{\text{first term}} = \frac{\frac{1}{15}}{\frac{1}{5}} = \frac{1}{3} ].
Since ( |r| = \frac{1}{3} < 1 ), the series is convergent.
Step 2
2.1.2 Calculate the sum to infinity of this series.
Answer
The formula for the sum ( S ) of an infinite geometric series is given by:
[ S = \frac{a}{1 - r} ]
Substituting the values:
[ S = \frac{\frac{1}{5}}{1 - \frac{1}{3}} = \frac{\frac{1}{5}}{\frac{2}{3}} = \frac{1}{5} \times \frac{3}{2} = \frac{3}{10} ].
Step 3
2.2.1 Write down the next TWO terms of the pattern.
Answer
The pattern of the sequence alternates between arithmetic and geometric components. Following the existing terms: ( P_0 = \frac{1}{3}, P_1 = 2, P_2 = 2 \cdots, P_3 = \frac{1}{27} ), we can find the next two terms:
- The next term after ( P_3 ) should continue the arithmetic part, resulting in ( P_4 = \frac{4}{3} )
- The term after that in the geometric sequence gives ( P_5 = \frac{2}{81} ).
Step 4
2.2.2 Determine the general term t_n for the odd terms of this pattern. Write down your answer in terms of x.
Answer
The odd terms of the sequence can be represented in general as follows: [ t_n = x + (n - 1) \cdot x ] This means every odd term can be expressed in terms of ( x ) where ( n ) is the term index.
Step 5
Step 6
2.2.4 If \( \sum_{n=1}^{21} P_n = 33.5 \), determine the value of x.
Answer
To find the value of ( x ), we set up the equation using the sum of the arithmetic and geometric series:
[ S = S_1 + S_0 = \frac{11}{2}(2x + 10) ]
This simplifies into the equation:
[ 33.5 = 66 + 0.5 ]
Solving for ( x ) yields ( x = 2.5 )