4. (a) Find the first 10 terms of the sequence, defined as follows; the first two terms are 1 and 3 respectively, each later term is formed by multiplying its predecessor by 3 and subtracting the next previous term.
(b) Evaluate the following
(i) [tex]$\sum_{r=1}^5 r^3$[/tex]
(ii) [tex]$\sum_{r=3}^7 3^r$[/tex]
Answer (2)
The first 10 terms of the sequence are 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765. The sum of the cubes of the first 5 natural numbers is 225, and the sum of the geometric series is 3267.
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Find the first 10 terms of the sequence using the recurrence relation: 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765.
Evaluate the sum of the first 5 cubes using the formula: ∑ r = 1 5 r 3 = 225 .
Evaluate the sum of the geometric series using the formula: ∑ r = 3 7 3 r = 3267 .
The first 10 terms are 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765; the sum of the first 5 cubes is 225; and the sum of the geometric series is 3267. $\boxed{225, 3267}
Explanation
Finding the First 10 Terms of the Sequence We are given a sequence where the first two terms are 1 and 3, and each subsequent term is generated by multiplying the previous term by 3 and subtracting the term before that. We need to find the first 10 terms of this sequence.
Calculating the Sequence Terms Let's denote the sequence as a n . We have a 1 = 1 and a 2 = 3 . The recurrence relation is given by a n = 3 a n − 1 − a n − 2 for n ≥ 3 . Now, we can find the next terms:
a 3 = 3 a 2 − a 1 = 3 ( 3 ) − 1 = 9 − 1 = 8 a 4 = 3 a 3 − a 2 = 3 ( 8 ) − 3 = 24 − 3 = 21 a 5 = 3 a 4 − a 3 = 3 ( 21 ) − 8 = 63 − 8 = 55 a 6 = 3 a 5 − a 4 = 3 ( 55 ) − 21 = 165 − 21 = 144 a 7 = 3 a 6 − a 5 = 3 ( 144 ) − 55 = 432 − 55 = 377 a 8 = 3 a 7 − a 6 = 3 ( 377 ) − 144 = 1131 − 144 = 987 a 9 = 3 a 8 − a 7 = 3 ( 987 ) − 377 = 2961 − 377 = 2584 a 10 = 3 a 9 − a 8 = 3 ( 2584 ) − 987 = 7752 − 987 = 6765
So, the first 10 terms of the sequence are: 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765.
Evaluating the Sum of Cubes Next, we need to evaluate the sum ∑ r = 1 5 r 3 . This is the sum of the cubes of the first 5 natural numbers. We can use the formula for the sum of the first n cubes, which is given by: r = 1 ∑ n r 3 = ( 2 n ( n + 1 ) ) 2 Substituting n = 5 into the formula, we get: r = 1 ∑ 5 r 3 = ( 2 5 ( 5 + 1 ) ) 2 = ( 2 5 ( 6 ) ) 2 = ( 2 30 ) 2 = ( 15 ) 2 = 225 Therefore, the sum of the cubes of the first 5 natural numbers is 225.
Evaluating the Geometric Series Now, we need to evaluate the sum ∑ r = 3 7 3 r . This is a geometric series with the first term 3 3 , common ratio 3, and 5 terms. The formula for the sum of a geometric series is: S n = r − 1 a ( r n − 1 ) where a is the first term, r is the common ratio, and n is the number of terms. In this case, a = 3 3 = 27 , r = 3 , and n = 5 . Substituting these values into the formula, we get: S 5 = 3 − 1 3 3 ( 3 5 − 1 ) = 2 27 ( 243 − 1 ) = 2 27 ( 242 ) = 27 ( 121 ) = 3267 Therefore, the sum of the geometric series is 3267.
Final Answer (a) The first 10 terms of the sequence are: 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765. (b) (i) ∑ r = 1 5 r 3 = 225 (ii) ∑ r = 3 7 3 r = 3267
Examples
Understanding sequences and series is fundamental in many areas of mathematics and real-world applications. For instance, the Fibonacci sequence (a sequence where each term is the sum of the two preceding ones) appears in nature, such as in the arrangement of leaves on a stem or the spirals of a sunflower. Geometric series are used in finance to calculate the future value of an investment or the present value of a loan. By mastering these concepts, you can model and analyze various phenomena, from population growth to compound interest.
