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Answer :
Sure, let's break down the solution to the question step-by-step:
(a) How far does he run on the 15th day?
The distances the athlete runs each day form an arithmetic sequence.
- The first term (a1) is 600 metres.
- The common difference (d) is 900 - 600 = 300 metres.
To find the distance run on the 15th day (a15), use the formula for the nth term of an arithmetic sequence:
[tex]\[ a_n = a_1 + (n-1) \times d \][/tex]
Where:
- [tex]\( a_1 = 600 \)[/tex]
- [tex]\( n = 15 \)[/tex]
- [tex]\( d = 300 \)[/tex]
[tex]\[
a_{15} = 600 + (15-1) \times 300 = 600 + 14 \times 300 = 600 + 4200 = 4800 \text{ metres}
\][/tex]
So, the athlete runs 4800 metres on the 15th day.
(b) What is the total distance he will run in 15 days?
To find the total distance run over 15 days, use the formula for the sum of the first n terms of an arithmetic sequence:
[tex]\[ S_n = \frac{n}{2} \times (a_1 + a_n) \][/tex]
Where:
- [tex]\( n = 15 \)[/tex]
- [tex]\( a_1 = 600 \)[/tex]
- [tex]\( a_n = 4800 \)[/tex] (calculated from part a)
[tex]\[
S_{15} = \frac{15}{2} \times (600 + 4800) = \frac{15}{2} \times 5400 = 7.5 \times 5400 = 40500 \text{ metres}
\][/tex]
Thus, the total distance run in 15 days is 40,500 metres.
(c) How long will it be before he can run a marathon of 42 km?
The total distance needed is 42 kilometres, which is 42,000 metres.
To find out how many days it will take to reach at least 42,000 metres, we add the terms of the sequence until their sum reaches or exceeds 42,000 metres. We have to find the smallest day [tex]\( n \)[/tex] such that:
[tex]\[ S_n = \frac{n}{2} \times (2a_1 + (n-1) \times d) \geq 42000 \][/tex]
By calculating the sequence and the total distance, we find:
On the 16th day, the athlete will finally surpass 42,000 metres.
Therefore, it will take 16 days before he can run a total of 42 km.
(a) How far does he run on the 15th day?
The distances the athlete runs each day form an arithmetic sequence.
- The first term (a1) is 600 metres.
- The common difference (d) is 900 - 600 = 300 metres.
To find the distance run on the 15th day (a15), use the formula for the nth term of an arithmetic sequence:
[tex]\[ a_n = a_1 + (n-1) \times d \][/tex]
Where:
- [tex]\( a_1 = 600 \)[/tex]
- [tex]\( n = 15 \)[/tex]
- [tex]\( d = 300 \)[/tex]
[tex]\[
a_{15} = 600 + (15-1) \times 300 = 600 + 14 \times 300 = 600 + 4200 = 4800 \text{ metres}
\][/tex]
So, the athlete runs 4800 metres on the 15th day.
(b) What is the total distance he will run in 15 days?
To find the total distance run over 15 days, use the formula for the sum of the first n terms of an arithmetic sequence:
[tex]\[ S_n = \frac{n}{2} \times (a_1 + a_n) \][/tex]
Where:
- [tex]\( n = 15 \)[/tex]
- [tex]\( a_1 = 600 \)[/tex]
- [tex]\( a_n = 4800 \)[/tex] (calculated from part a)
[tex]\[
S_{15} = \frac{15}{2} \times (600 + 4800) = \frac{15}{2} \times 5400 = 7.5 \times 5400 = 40500 \text{ metres}
\][/tex]
Thus, the total distance run in 15 days is 40,500 metres.
(c) How long will it be before he can run a marathon of 42 km?
The total distance needed is 42 kilometres, which is 42,000 metres.
To find out how many days it will take to reach at least 42,000 metres, we add the terms of the sequence until their sum reaches or exceeds 42,000 metres. We have to find the smallest day [tex]\( n \)[/tex] such that:
[tex]\[ S_n = \frac{n}{2} \times (2a_1 + (n-1) \times d) \geq 42000 \][/tex]
By calculating the sequence and the total distance, we find:
On the 16th day, the athlete will finally surpass 42,000 metres.
Therefore, it will take 16 days before he can run a total of 42 km.
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