Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2007 | January | Q#9

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Question

Ann has some sticks that are all of the same length. She arranges them in squares  and has made the following 3 rows of patterns:

She notices that 4 sticks are required to make the single square in the first row, 7  sticks to make 2 squares in the second row and in the third row she needs 10 sticks  to make 3 squares.

a.   Find an expression, in terms of n, for the number of sticks required to make a  similar arrangement of n squares in the nth row.

Ann continues to make squares following the same pattern. She makes 4 squares in  the 4th row and so on until she has completed 10 rows.

b.   Find the total number of sticks Ann uses in making these 10 rows. 

Ann started with 1750 sticks. Given that Ann continues the pattern to complete k  rows but does not have sufficient sticks to complete the (k + 1)th row,

c.   show that k satisfies (3k – 100)(k + 35) < 0.

d.   Find the value of k.

Solution

a.
 

We are given that for the first row Ann requires 4 sticks, for the 2nd row she requires  7 sticks, for the 3rd row she requires 10 sticks and so on. 

It is evident that these requirements for each next row constitute an arithmetic series  as follows.

For row 1, square 1;                                                   

For row 2, squares 2;                                                 

For row 3, squares 3;                                                 

Expression for difference  in Arithmetic Progression (A.P) is:

Therefore;

To find an expression for nth row what is the number of sticks required for n squares,  we need expression for general term of an arithmetic series.

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

Therefore;

b.
 

We are required to find the total number of sticks used by Ann to make 10 rows of a  similar pattern.

We need to find the sum of all the sticks used in all 10 rows of the pattern.

As seen in (a), number of sticks used make an arithmetic series;

We need to find sum of first 10 such terms.

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

c.
 

We are given that Ann had 1750 sticks and completed the pattern till kth row but  could not complete (k+1)th row.

Therefore, it can be inferred that total sticks required to complete k rows must be  less than 1750.

As we have demonstrated in (b) that number of sticks used make n rows is given by; 

As we have discussed above ;

d.
 

To find the value of k we need to solve the inequality;

We solve the following equation to find critical values of ;

As demonstrated in (c), we know that it can be written as; 

Now we have two options;

Hence the critical points on the curve for the given condition are  & -35. 

Since k represents number of rows it cannot be negative and hence only plausible value for k is .

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