Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2006 | January | Q#7

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Question

On Alice’s 11th birthday she started to receive an annual allowance. The first annual allowance was  £500 and on each following birthday the allowance was increased by £200.

a.   Show that, immediately after her 12th birthday, the total of the allowances that Alice had received  was £1200.

b.   Find the amount of Alice’s annual allowance on her 18th birthday.

c.   Find the total of the allowances that Alice had received up to and including her 18th birthday. 

When the total of the allowances that Alice had received reached £32 000 the allowance stopped. 

d.   Find how old Alice was when she received her last allowance.

Solution

a.
 

1st Allowance Received= 11th Birthday Allowance=500

2nd Allowance Received= 12th Birthday Allowance=500+(500+200)=1200

Hence, total amount of allowance immediately after 13th birthday will be £1200.

Alternate Method

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

Therefore;

b.
 

We are required to find the amount of Alice’s annual allowance on her 18th birthday.

It is evident that allowance received on 11th birthday is first term in this arithmetic series and  allowance received on 18th birthday is 8th term.

We are required to find 8th term.

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

Therefore;

18th Birthday Allowance=£1900

c.
 

We are required to find the total of the allowances that Alice had received up to and including her  18th birthday. It is evident from (b) that allowance received on 18th birthday is 8th allowance  received. 

Therefore;

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

Total Amount Received Till 18th
Birthday=
£9600

d.
 

When the total of the allowances that Alice had received reached £32000 the allowance stopped.  We are required to find how old Alice was when she received her last allowance. 

We first need to find the total number terms which sum up to £32000.

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

Now we have two options.

Since total number of terms cannot be negative, only possible total number of terms is 16. 

First term of allowance was received at the age of 11 years and remaining 15 of the total 16 terms  were received later.

Hence, age of Alice at the time of last received allowance is 26 years.

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