Summation of series — AS & A Level Further Mathematics (9231) questions by topic
Series techniques converge in this topic: showing that a sum of rectangle areas bounds a definite integral of 2x - x^2, using the method of differences on a fraction involving k and r, and proving a series identity for 1 + 2x + 3x^2 and so on by mathematical induction.
Find, show, deduce, state and express are the top command words across 28 questions from 2023 to 2025 papers, worth 6 to 15 marks, where the inductive step and the differences telescoping correctly are what earn most of the credit. With no examiner-report extracts logged for this topic, submitting each induction or method-of-differences proof for instant marking is the way to confirm every algebraic step is justified before the final result is trusted.
May–June 2025, Paper 11
Use standard results from the list of formulae (MF19) to show that sum{r=1}^{n} (2 - 3r)(5 - 3r) = an^3 + bn^2 + cn, where a, b and c are integers to be determined.
Answer this question and get it marked →May–June 2025, Paper 12
Use standard results from the list of formulae (MF19) to show that sum{r=1}^{n} (2 - 3r)(5 - 3r) = an^3 + bn^2 + cn, where a, b and c are integers to be determined.
Answer this question and get it marked →May–June 2025, Paper 13
Let wr = r(r+1)(r+2)...(r+9). Show that w{r+1} - wr = 10(r+1)(r+2)...(r+9).
Answer this question and get it marked →May–June 2025, Paper 21
The diagram shows the curve with equation y = (1/sqrt(x)) e^(sqrt(x)) for x = 1, together with a set of n - 1 rectangles of unit width. The rectangles are drawn between x = 1 and…
Answer this question and get it marked →May–June 2025, Paper 23
The diagram shows the curve with equation y = 1/(x^2 + 1) for 0 <= x <= 1, together with a set of n rectangles of width 1/n. The rectangles are drawn on the subintervals between…
Answer this question and get it marked →May–June 2024, Paper 12
Use standard results from the list of formulae (MF19) to show that sum{r=1}^{N} r(r+1)(3r+4) = (1/12)N(N+1)(N+2)(9N+19).
Answer this question and get it marked →May–June 2024, Paper 13
Prove by mathematical induction that, for all positive integers n, sum{r=1}^{n} r^2 = (1/6) n(n+1)(2n+1).
Answer this question and get it marked →May–June 2024, Paper 21
The diagram shows the curve with equation y = 2x - x^2 for 0 <= x <= 1, together with a set of n rectangles of width 1/n. By considering the sum of the areas of these rectangles,…
Answer this question and get it marked →May–June 2024, Paper 22
The diagram shows the curve with equation y = 2x - x^2 for 0 <= x <= 1, together with a set of n rectangles of width 1/n. By considering the sum of the areas of these rectangles,…
Answer this question and get it marked →May–June 2024, Paper 23
The diagram shows the curve with equation y = x^(-2) for 2 <= x <= N together with a set of (N - 2) rectangles of unit width. By considering the sum of the areas of these…
Answer this question and get it marked →Show that the sum from r = 1 to n of z^(4r) = (z^(4n+2) - z^2) / (z^2 - z^(-2)), for z^2 not equal to z^(-2).
Answer this question and get it marked →October–November 2024, Paper 11
Use the method of differences to find sum{r=1}^{n} 5k/((5r + k)(5r + 5 + k)) in terms of n and k, where k is a positive constant.
Answer this question and get it marked →October–November 2024, Paper 12
It is given that Sn = sum from r = 1 to n of ur, where ur = x^(f(r)) - x^(f(r+1)) and x 0. Find Sn in terms of n, x and the function f.
Answer this question and get it marked →October–November 2024, Paper 13
Use the method of differences to find sum{r=1}^{n} 5k/((5r + k)(5r + 5 + k)) in terms of n and k, where k is a positive constant.
Answer this question and get it marked →October–November 2024, Paper 21
The diagram shows the curve with equation y = (1/2)^x for 0 <= x <= 1, together with a set of N rectangles each of width 1/N. By considering the sum of the areas of these…
Answer this question and get it marked →October–November 2024, Paper 22
The diagram shows the curve with equation y = e^(1 - x) for 0 <= x <= 1, together with a set of n rectangles of width 1/n. By considering the sum of the areas of these rectangles,…
Answer this question and get it marked →October–November 2024, Paper 23
The diagram shows the curve with equation y = (1/2)^x for 0 <= x <= 1, together with a set of N rectangles each of width 1/N. By considering the sum of the areas of these…
Answer this question and get it marked →May–June 2023, Paper 11
Use the method of differences to find sum{r=1}^{n} 1/((kr + 1)(kr - k + 1)) in terms of n and k, where k is a positive constant.
Answer this question and get it marked →May–June 2023, Paper 12
Use the method of differences to find sum{r=1}^{n} 1/( (kr + 1)(kr - k + 1) ) in terms of n and k, where k is a positive constant.
Answer this question and get it marked →May–June 2023, Paper 13
Use standard results from the list of formulae (MF19) to show that sum{r=1}^{n} (6r^2 + 6r - 5) = an^3 + bn^2 + cn, where a, b and c are integers to be determined.
Answer this question and get it marked →May–June 2023, Paper 23
The diagram shows the curve with equation y = (1 - x)^2 for 0 <= x <= 1, together with a set of n rectangles of width 1/n. By considering the sum of the areas of these rectangles,…
Answer this question and get it marked →October–November 2023, Paper 11
By considering (r + 1)^2 - r^2, use the method of differences to prove that sum{r=1}^{n} r = (1/2) n (n + 1).
Answer this question and get it marked →Prove by mathematical induction that, for all positive integers n, 1 + 2x + 3x^2 + ... + n x^(n-1) = (1 - (n + 1) x^n + n x^(n+1)) / (1 - x)^2.
Answer this question and get it marked →October–November 2023, Paper 12
Use standard results from the list of formulae (MF19) to find sum{r=1}^{n} (3r^2 + 3r + 1), simplifying your answer.
Answer this question and get it marked →October–November 2023, Paper 13
By considering (r+1)^2 - r^2, use the method of differences to prove that sum{r=1}^{n} r = (1/2) n(n+1).
Answer this question and get it marked →Prove by mathematical induction that, for all positive integers n, 1 + 2x + 3x^2 + ... + nx^(n-1) = (1 - (n+1)x^n + nx^(n+1)) / (1 - x)^2.
Answer this question and get it marked →October–November 2023, Paper 21
State the sum of the series 1 + z + z^2 + ... + z^(n-1), for z != 1.
Answer this question and get it marked →October–November 2023, Paper 23
State the sum of the series 1 + z + z^2 + ... + z^(n-1), for z != 1.
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