Binomial expansion and sequences – Year 12 A level maths worksheet with answers to download as PDF

Use the binomial expansion.

1. Expand \((2+3x)^4\) in ascending powers of \(x\). [4 marks]
2. Find the coefficient of \(x^3\) in the expansion of \((1-2x)^7\). [3 marks]
3. Evaluate \(\binom{10}3\) and \(\binom85\). [2 marks]

Find the required term in each expansion.

1. The term independent of \(x\) in \(\left(x^2+\dfrac1x\right)^9\). [4 marks]
2. The constant term in \(\left(x+\dfrac2x\right)^6\). [4 marks]
3. The coefficient of \(x^4\) in \((3-x)^6\). [3 marks]

Use the first three terms of a binomial expansion to estimate each value.

1. Write down the first three terms of \((1+x)^8\) and use \(x=0.02\) to estimate \(1.02^8\). [3 marks]
2. Expand \((2-x)^6\) up to and including the term in \(x^3\). [4 marks]
3. Use your answer to part (b) to estimate \(1.98^6\) and compare with the calculator value. [3 marks]

An arithmetic sequence begins 5, 9, 13, 17, \dots

1. Find the 20th term. [2 marks]
2. Find the sum of the first 20 terms. [2 marks]
3. Find the first term greater than 200. [3 marks]
4. Find the least \(n\) for which the sum of the first \(n\) terms exceeds 1000. [4 marks]

A geometric sequence has first term 3 and common ratio 2.

1. Find the 8th term. [2 marks]
2. Find the sum of the first 10 terms. [2 marks]
3. A different geometric series has first term 64 and common ratio \(-\tfrac12\). Show that it converges and find its sum to infinity. [3 marks]
4. Find the sum to infinity of a series whose first term is 12 and whose second term is 8. [3 marks]

Work with sequences defined in different ways.

1. Evaluate \(\displaystyle\sum_{r=1}^{20}(3r+2)\). [3 marks]
2. A sequence has \(u_{n+1}=2u_n-3\) and \(u_1=5\). Find \(u_2\), \(u_3\) and \(u_4\). [3 marks]
3. A sequence has \(u_1=2\) and \(u_{n+1}=\dfrac1{1-u_n}\). Show that it has period 3 and find \(\displaystyle\sum_{n=1}^{100}u_n\). [5 marks]

Priya starts a savings plan. She pays in £50 in the first month and increases her deposit by £5 every month.

1. How much does she deposit in month 24? [2 marks]
2. How much has she saved in total after 24 months? [2 marks]
3. Show that the total after \(n\) months is \(\tfrac n2(95+5n)\). [2 marks]
4. After how many months does her total first exceed £5 000? [4 marks]

£2 000 is invested at 3.5% compound interest per year. The value after \(n\) years is \(V_n=2000\times1.035^n\).

1. Check the value after 2 years and 3 years. [2 marks]
2. Find the value after 10 years. [2 marks]
3. After how many whole years does the investment first exceed £4 000? [4 marks]
4. Identify the type of sequence and the common ratio. [2 marks]

The first row of a stadium stand has 20 seats. Each further row has 3 more seats than the row in front.

1. How many seats are in row 30? [2 marks]
2. How many seats are in the first 30 rows? [2 marks]
3. A concert needs 1 500 seats. What is the smallest number of rows needed? [5 marks]
4. Tickets are £35. How much would a full 30-row stand raise? [1 mark]

A landlord raises the monthly rent by 3% every year. The rent is currently £800. After \(n\) years the rent is \(800\times1.03^n\).

1. Write down the first three terms of the expansion of \((1+0.03)^6\) and use it to estimate the rent after 6 years. [4 marks]
2. Find the exact rent after 6 years to the nearest penny. [1 mark]
3. Find the percentage error in the estimate. [2 marks]
4. Explain why the estimate is slightly low. [1 mark]

A ball is dropped from a height of 2 m. After each bounce it rises to 70% of its previous height.

1. Find the height reached after the 5th bounce. [2 marks]
2. Find the total distance travelled by the ball before it stops bouncing. [5 marks]
3. After how many bounces is the rebound height first less than 1 cm? [4 marks]

The number of fish in a lake at the start of year \(n\) is \(P_n\). Each year 10% of the fish die and 50 new fish are added. So \(P_{n+1}=0.9P_n+50\) with \(P_0=400\).

1. Find \(P_1\), \(P_2\) and \(P_3\). [3 marks]
2. Find the limit \(L\) that the population approaches, by solving \(L=0.9L+50\). [2 marks]
3. Show that \(P_n=500-100\times0.9^n\) and check it for \(n=2\). [3 marks]
4. After how many years does the population first exceed 480? [3 marks]

Use the binomial expansion of \((3-2x)^5\).

1. Find the first three terms in ascending powers of \(x\). [4 marks]
2. Use \(x=0.01\) to estimate \(2.98^5\). [2 marks]
3. Find the error in your estimate to 3 decimal places. [2 marks]
4. Explain why taking more terms would improve the estimate. [1 mark]

A runner trains for a marathon. In week 1 she runs 8 km and each week she runs 1.5 km further than the week before.

1. How far does she run in week 12? [2 marks]
2. What is the total distance in the first 12 weeks? [2 marks]
3. In which week does she first run more than 30 km? [3 marks]
4. The race is 42.195 km. In which week does she first run further than the marathon distance? [3 marks]

Aisha is offered two jobs. Job A pays £24 000 in year 1 with a rise of £1 500 each year. Job B pays £22 000 in year 1 with a 5% rise each year.

1. Check the Job B salary in year 3 and identify the type of sequence for each job. [3 marks]
2. Find the total earned from Job A in the first 10 years. [2 marks]
3. Find the total earned from Job B in the first 10 years. [3 marks]
4. In which year does the Job B salary first exceed the Job A salary? [5 marks]

In the expansion of \((1+ax)^n\) in ascending powers of \(x\), the coefficient of \(x\) is \(-12\) and the coefficient of \(x^2\) is 60.

1. Show that \(na=-12\) and \(\tfrac12n(n-1)a^2=60\). [2 marks]
2. Find \(n\) and \(a\). [4 marks]
3. Find the coefficient of \(x^3\). [2 marks]
4. Use the expansion with \(x=0.01\) to estimate \(0.98^6\) to 5 decimal places. [3 marks]

A geometric series has all positive terms. The second term is 12 and the fourth term is 3.

1. Show that \(r=\tfrac12\) and find \(a\). [3 marks]
2. Find the sum of the first 8 terms. [2 marks]
3. Find the sum to infinity. [2 marks]
4. Find the sum of the terms from the 5th term onwards. [3 marks]

A sequence is defined by \(u_{n+1}=3u_n-4\) and \(u_1=k\).

1. Show that \(u_3=9k-16\). [2 marks]
2. Given that \(u_3=11\) find \(k\) and list the first four terms. [3 marks]
3. Find the value \(L\) for which \(u_{n+1}=u_n=L\) (a fixed point). [2 marks]
4. For \(k=3\) show that \(u_n=2+3^{n-1}\) and find \(\sum_{n=1}^{10}u_n\). [5 marks]

A water tank has capacity 24 litres. In the first minute 5 litres flow in. In each later minute the amount is 80% of the amount in the previous minute.

1. Check the totals for minutes 2 and 4. [2 marks]
2. Show that the total after \(n\) minutes is \(25(1-0.8^n)\). [2 marks]
3. Show that the tank would eventually fill, even though the flow keeps falling. [2 marks]
4. After how many minutes is the tank full? [4 marks]

Three numbers form an arithmetic sequence. Their sum is 21 and their product is 231.

1. Find the three numbers. [5 marks]
2. The sequence continues as 3, 7, 11, \dots Show that the sum of the first \(n\) terms is \(S_n=2n^2+n\). [3 marks]
3. Find \(n\) if \(S_n=1275\). [3 marks]
4. Using \(u_n=S_n-S_{n-1}\) verify that \(u_n=4n-1\). [3 marks]