Sue is training for a marathon. Her training includes a run every Saturday starting with a run of 5 km on the first Saturday. Each Saturday she increases the length of her run from the previous Saturday by 2 km. (a) Show that on the 4th Saturday of training she runs 11 km. (b) Find an expression, in terms of n, for the length of her training run on the nth Saturday. (c) Show that the total distance she runs on Saturdays in n weeks of training is n(n + 4) km. On the nth Saturday Sue runs 43 km. (d) Find the value of n. (e) Find the total distance, in km, Sue runs on Saturdays in n weeks of training.

A-Level Edexcel Maths Pure Equation of a Straight Line: Sue is training for a marathon.

Sue is training for a marathon - Edexcel - A-Level Maths Pure - Question 9 - 2008 - Paper 1

Question 9

Sue is training for a marathon. Her training includes a run every Saturday starting with a run of 5 km on the first Saturday. Each Saturday she increases the length ... show full transcript

Worked Solution & Example Answer:Sue is training for a marathon - Edexcel - A-Level Maths Pure - Question 9 - 2008 - Paper 1

Step 1

Show that on the 4th Saturday of training she runs 11 km.

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Answer

Sue starts her training with 5 km on the first Saturday, and each subsequent Saturday she increases her distance by 2 km. Therefore, the distances for the first four Saturdays are:

1st Saturday: $5$ km
2nd Saturday: $5 + 2 = 7$ km
3rd Saturday: $7 + 2 = 9$ km
4th Saturday: $9 + 2 = 11$ km

Thus, she runs 11 km on the 4th Saturday.

Step 2

Find an expression, in terms of n, for the length of her training run on the nth Saturday.

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Answer

The distance run on the nth Saturday can be expressed as: $t_n = 5 + 2(n - 1)$ This expression represents the 5 km on the first Saturday plus an increase of 2 km for each additional Saturday.

Step 3

Show that the total distance she runs on Saturdays in n weeks of training is n(n + 4) km.

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Answer

The distance for each Saturday can be summed as: $S_n = rac{n}{2} imes (t_1 + t_n)$ Where $t_1 = 5$ and $t_n = 5 + 2(n - 1)$ gives us: $t_n = 5 + 2n - 2 = 2n + 3$ So, the total distance is: $S_n = rac{n}{2} imes (5 + (2n + 3)) = rac{n}{2} imes (2n + 8) = n(n + 4)$

Step 4

Find the value of n.

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Answer

We know that on the nth Saturday, Sue runs 43 km: $2n + 3 = 43$ Solving this equation: $2n = 43 - 3 = 40$ $n = 20$ Thus, the value of n is 20.

Step 5

Find the total distance, in km, Sue runs on Saturdays in n weeks of training.

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Answer

Using the formula derived earlier, when n = 20: $S_{20} = 20(20 + 4) = 20 imes 24 = 480 ext{ km}$ Thus, the total distance she runs is 480 km.

Sue is training for a marathon - Edexcel - A-Level Maths Pure - Question 9 - 2008 - Paper 1

Worked Solution & Example Answer:Sue is training for a marathon - Edexcel - A-Level Maths Pure - Question 9 - 2008 - Paper 1

Show that on the 4th Saturday of training she runs 11 km.

Find an expression, in terms of n, for the length of her training run on the nth Saturday.

Show that the total distance she runs on Saturdays in n weeks of training is n(n + 4) km.

Find the value of n.

Find the total distance, in km, Sue runs on Saturdays in n weeks of training.

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