1.04h Arithmetic sequences: nth term and sum formulae

The first term of an arithmetic progression is 12 and the sum of the first 9 terms is 135.

1. Find the common difference of the progression. [2]

The first term, the ninth term and the \(n\)th term of this arithmetic progression are the first term, the second term and the third term respectively of a geometric progression.

1. Find the common ratio of the geometric progression and the value of \(n\). [5]

The function f is defined for \(x \geqslant 0\). It is given that f has a minimum value when \(x = 2\) and that \(\text{f}''(x) = (4x + 1)^{-\frac{1}{2}}\).

1. Find \(\text{f}'(x)\). [3]

It is now given that \(\text{f}''(0)\), \(\text{f}'(0)\) and \(\text{f}(0)\) are the first three terms respectively of an arithmetic progression.

1. Find the value of \(\text{f}(0)\). [3]
2. Find \(\text{f}(x)\), and hence find the minimum value of f. [5]

Two heavyweight boxers decide that they would be more successful if they competed in a lower weight class. For each boxer this would require a total weight loss of 13 kg. At the end of week 1 they have each recorded a weight loss of 1 kg and they both find that in each of the following weeks their weight loss is slightly less than the week before. Boxer A's weight loss in week 2 is 0.98 kg. It is given that his weekly weight loss follows an arithmetic progression.

1. Write down an expression for his total weight loss after \(x\) weeks. [1]
2. He reaches his 13 kg target during week \(n\). Use your answer to part (i) to find the value of \(n\). [2]

Boxer B's weight loss in week 2 is 0.92 kg and it is given that his weekly weight loss follows a geometric progression.

1. Calculate his total weight loss after 20 weeks and show that he can never reach his target. [4]

The first and second terms of a progression are 4 and 8 respectively. Find the sum of the first 10 terms given that the progression is

1. an arithmetic progression, [2]
2. a geometric progression. [2]

Three geometric progressions, \(P\), \(Q\) and \(R\), are such that their sums to infinity are the first three terms respectively of an arithmetic progression. Progression \(P\) is \(2, 1, \frac{1}{2}, \frac{1}{4}, \ldots\) Progression \(Q\) is \(3, 1, \frac{1}{3}, \frac{1}{9}, \ldots\)

1. Find the sum to infinity of progression \(R\). [3]
2. Given that the first term of \(R\) is 4, find the sum of the first three terms of \(R\). [3]

The first three terms of an arithmetic progression are \(4\), \(x\) and \(y\) respectively. The first three terms of a geometric progression are \(x\), \(y\) and \(18\) respectively. It is given that both \(x\) and \(y\) are positive.

1. Find the value of \(x\) and the value of \(y\). [4]
2. Find the fourth term of each progression. [3]

The \(r\)th term of an arithmetic series is \((2r - 5)\).

1. Write down the first three terms of this series. [2]
2. State the value of the common difference. [1]
3. Show that \(\sum_{r=1}^n (2r - 5) = n(n - 4)\). [3]

An arithmetic series has first term \(a\) and common difference \(d\).

1. Prove that the sum of the first \(n\) terms of the series is $$\frac{1}{2}n[2a + (n - 1)d].$$ [4]

Sean repays a loan over a period of \(n\) months. His monthly repayments form an arithmetic sequence. He repays £149 in the first month, £147 in the second month, £145 in the third month, and so on. He makes his final repayment in the \(n\)th month, where \(n > 21\).

1. Find the amount Sean repays in the 21st month. [2]

Over the \(n\) months, he repays a total of £5000.

1. Form an equation in \(n\), and show that your equation may be written as $$n^2 - 150n + 5000 = 0.$$ [3]
2. Solve the equation in part (c). [3]
3. State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem. [1]

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.

1. Show that, immediately after her 12th birthday, the total of the allowances that Alice had received was £1200. [1]
2. Find the amount of Alice's annual allowance on her 18th birthday. [2]
3. Find the total of the allowances that Alice had received up to and including her 18th birthday. [3]

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

1. Find how old Alice was when she received her last allowance. [7]

An athlete prepares for a race by completing a practice run on each of 11 consecutive days. On each day after the first day he runs further than he ran on the previous day. The lengths of his 11 practice runs form an arithmetic sequence with first term \(a\) km and common difference \(d\) km. He runs 9 km on the 11th day, and he runs a total of 77 km over the 11 day period. Find the value of \(a\) and the value of \(d\). [7]

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: Row 1 \(\square\) Row 2 \(\square\square\) Row 3 \(\square\square\square\) 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.

1. Find an expression, in terms of \(n\), for the number of sticks required to make a similar arrangement of \(n\) squares in the \(n\)th row. [3]

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.

1. Find the total number of sticks Ann uses in making these 10 rows. [3]

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,

1. show that \(k\) satisfies \((3k - 100)(k + 35) < 0\). [4]
2. Find the value of \(k\). [2]

Each year, for 40 years, Anne will pay money into a savings scheme. In the first year she pays £500. Her payments then increase by £50 each year, so that she pays £550 in the second year, £600 in the third year, and so on.

1. Find the amount that Anne will pay in the 40th year. [2]
2. Find the total amount that Anne will pay in over the 40 years. [2]

Over the same 40 years, Brian will also pay money into the savings scheme. In the first year he pays in £890 and his payments then increase by £\(d\) each year. Given that Brian and Anne will pay in exactly the same amount over the 40 years,

1. find the value of \(d\). [4]

1. An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum of the first \(n\) terms of the series is $$\frac{1}{2}n[2a + (n-1)d].$$ [4]

A company made a profit of £54 000 in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference £\(d\). This model predicts total profits of £619 200 for the 9 years 2001 to 2009 inclusive.

1. Find the value of \(d\). [4]

Using your value of \(d\),

1. find the predicted profit for the year 2011. [2]

In the first month after opening, a mobile phone shop sold 280 phones. A model for future trading assumes that sales will increase by \(x\) phones per month for the next 35 months, so that \((280 + x)\) phones will be sold in the second month, \((280 + 2x)\) in the third month, and so on. Using this model with \(x = 5\), calculate

1. 1. the number of phones sold in the 36th month, [2]
   2. the total number of phones sold over the 36 months. [2]

The shop sets a sales target of 17 000 phones to be sold over the 36 months. Using the same model,

1. find the least value of \(x\) required to achieve this target. [4]

The sum of an arithmetic series is $$\sum_{r=1}^{n} (80 - 3r).$$

1. Write down the first two terms of the series. [2]
2. Find the common difference of the series. [1]

Given that \(n = 50\),

1. find the sum of the series. [3]

The first three terms of an arithmetic series are \(p\), \(5p - 8\), and \(3p + 8\) respectively.

1. Show that \(p = 4\). [2]
2. Find the value of the 40th term of this series. [3]
3. Prove that the sum of the first \(n\) terms of the series is a perfect square. [3]