Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2016 | June | Q#9

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Question

On John’s 10th birthday he received the first of an annual birthday gift of money from his uncle. This  first gift was £60 and on each subsequent birthday the gift was £15 more than the year before.  The amounts of these gifts form an arithmetic sequence.

a.   Show that, immediately after his 12th birthday, the total of these gifts was £225.

b.   Find the amount that John received from his uncle as a birthday gift on his 18th birthday.

c.   Find the total of these birthday gifts that John had received from his uncle up to and including his  21st birthday.

When John had received n of these birthday gifts, the total money that he had received from these  gifts was £3375

d.   Show that n2 + 7n = 25 × 18

e.   Find the value of n, when he had received £3375 in total, and so determine John’s age at this time.

Solution

a.
 

It is given that the amounts of annual birthday gift received by John form an arithmetic sequence.

From the given information we can collect following data about the said arithmetic sequence.

We are looking for the sum of amounts received on 10th, 11th and 12th birthdays (sum of first 03  terms of the said arithmetic sequence).

Expression for the sum of  number of terms in the Arithmetic Progression (A.P) is:

Therefore;

Hence, immediately after his 12th birthday, the total of these gifts was £225.

b.
 

We are required to find the amount that John received from his uncle as a birthday gift on his 18th  birthday.

It is evident that we are looking for 9th term of the said arithmetic sequence.

Expression for the general term  in the Arithmetic Progression (A.P) is:

Therefore;

Hence, the amount that John received from his uncle as a birthday gift on his 18th birthday is £180.

c.
 

We are required to find the total of these birthday gifts that John had received from his uncle up to  and including his 21st birthday.

It is evident that we are looking for the sum of arithmetic series from first term 12 terms (from 10th  birthday to 21st).

Expression for the sum of  number of terms in the Arithmetic Progression (A.P) is:

Therefore;

Hence, total of these birthday gifts that John had received from his uncle up to and including his  21st birthday, is £1710.

d.
 

We are given that total of n birthday gifts that John had received from his uncle is £3375. 

Expression for the sum of  number of terms in the Arithmetic Progression (A.P) is:

Therefore;

e.
 

We have found in (d) that;

Now we have two options.

Since, number of birthday cannot be negative, the total amount of gifts £3375 was received on 18th  year since 10th birthday of John.

Therefore, age of John at that time would be 9+18=27 years.

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