Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2006  January  Q#7
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Question
On Alice’s 11^{th} 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 18^{th} 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.
1^{st }Allowance Received= 11^{th} Birthday Allowance=500
2^{nd} Allowance Received= 12^{th }Birthday Allowance=500+(500+200)=1200
Hence, total amount of allowance immediately after 13^{th} 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 18^{th} birthday.
It is evident that allowance received on 11^{th }birthday is first term in this arithmetic series and allowance received on 18^{th} birthday is 8^{th} term.
We are required to find 8^{th} term.
Expression for the general term in the Arithmetic Progression (A.P) is:
Therefore;
18^{th} Birthday Allowance=£1900
c.
We are required to find the total of the allowances that Alice had received up to and including her 18^{th} birthday. It is evident from (b) that allowance received on 18^{th} birthday is 8^{th} allowance received.
Therefore;
Expression for the sum of number of terms in the Arithmetic Progression (A.P) is:
Total Amount Received Till 18^{th}
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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