The sequence \(u_1, u_2, u_3, \ldots\) is such that \(u_1 = 5\) and \(u_{n+1} = 6u_n + 5\) for \(n \geq 1\).
(a) Prove by induction that \(u_n = 6^n - 1\) for all positive integers \(n\).
(b) Deduce that \(u_{2n}\) is divisible by \(u_n\) for \(n \geq 1\).
The sequence \(u_1, u_2, u_3, \ldots\) is such that \(u_1 = 5\) and \(u_{n+1} = 6u_n + 5\) for \(n \geq 1\).
(a) Prove by induction that \(u_n = 6^n - 1\) for all positive integers \(n\).
(b) Deduce that \(u_{2n}\) is divisible by \(u_n\) for \(n \geq 1\).
Prove by mathematical induction that \(2025^n + 47^n - 2\) is divisible by 46 for all positive integers \(n\).
Prove by mathematical induction that \(6^{4n} + 38^n - 2\) is divisible by 74 for all positive integers \(n\). [6]
Prove by mathematical induction that \(6^{4n} + 38^n - 2\) is divisible by 74 for all positive integers \(n\).
Prove by mathematical induction that, for all positive integers n, \(5^{3n} + 32^n - 33\) is divisible by 31.
Prove by mathematical induction that, for all positive integers n, \(7^{2n} + 97^n - 50\) is divisible by 48. [6]
Prove by mathematical induction that \(2^{4n} + 3^{1n} - 2\) is divisible by 15 for all positive integers \(n\).
1 Prove by mathematical induction that \(3^{3 n}-1\) is divisible by 13 for every positive integer \(n\).
It is given that \(\mathrm{f}(n)=2^{3 n}+8^{n-1}\). By simplifying \(\mathrm{f}(k)+\mathrm{f}(k+1)\), or otherwise, prove by mathematical induction that \(\mathrm{f}(n)\) is divisible by 9 for every positive integer \(n\).
Prove, by mathematical induction, that \(5^{n}+3\) is divisible by 4 for all non-negative integers \(n\).
It is given that \(\phi(n)=5^{n}(4 n+1)-1\), for \(n=1,2,3, \ldots\). Prove, by mathematical induction, that \(\phi(n)\) is divisible by 8 , for every positive integer \(n\).
Prove by mathematical induction that, for all non-negative integers \(n\),
\(11^{2 n}+25^{n}+22\)
is divisible by 24 .
Prove by mathematical induction that, for all positive integers \(n, 10^{n}+3 \times 4^{n+2}+5\) is divisible by 9 .
Prove by mathematical induction that \(5^{2 n}-1\) is divisible by 8 for every positive integer \(n\).
It is given that \(\mathrm{f}(n)=3^{3 n}+6^{n-1}\).
(i) Show that \(\mathrm{f}(n+1)+\mathrm{f}(n)=28\left(3^{3 n}\right)+7\left(6^{n-1}\right)\).
(ii) Hence, or otherwise, prove by mathematical induction that \(\mathrm{f}(n)\) is divisible by 7 for every positive integer \(n\).
Prove by mathematical induction that, for all non-negative integers \(n\),
\(11^{2 n}+25^{n}+22\)
is divisible by 24 .
Answer only one of the following two alternatives.
EITHER
(i) By considering \((2r+1)^2-(2r-1)^2\), use the method of differences to prove that
\(\sum_{r=1}^{n}r=\frac12n(n+1).\)
(ii) By considering \((2r+1)^4-(2r-1)^4\), use the method of differences and the result given in part (i) to prove that
\(\sum_{r=1}^{n}r^3=\frac14n^2(n+1)^2.\)
The sums \(S\) and \(T\) are defined as follows:
\(S=1^3+2^3+3^3+4^3+\cdots+(2N)^3+(2N+1)^3,\)
\(T=1^3+3^3+5^3+7^3+\cdots+(2N-1)^3+(2N+1)^3.\)
(iii) Use the result given in part (ii) to show that \(S=(2N+1)^2(N+1)^2\).
(iv) Hence, or otherwise, find an expression in terms of \(N\) for \(T\), factorising your answer as far as possible.
(v) Deduce the value of \(\dfrac ST\) as \(N\to\infty\).
OR
The curve \(C\) has equation \(x^2+2xy=y^3-2\).
(i) Show that \(A(-1,1)\) is the only point on \(C\) with \(x\)-coordinate equal to \(-1\).
For \(n\ge1\), let \(A_n\) denote the value of \(\dfrac{d^ny}{dx^n}\) at the point \(A(-1,1)\).
(ii) Show that \(A_1=0\).
(iii) Show that \(A_2=\dfrac25\).
Let \(I_n=\int_{-1}^{0}x^n\dfrac{d^ny}{dx^n}\,dx\).
(iv) Show that for \(n\ge2\),
\(I_n=(-1)^{n+1}A_{n-1}-nI_{n-1}.\)
(v) Deduce the value of \(I_3\) in terms of \(I_1\).
Solutions to this question by accurate drawing will not be accepted. A circle has centre \((4,2)\) and meets the \(x\)-axis at \((-2,0)\).
(a) Find the equation of the circle.
(b) Find, in exact form, the coordinates of the points where the circle meets the \(y\)-axis.
The point \(A\) has coordinates \((-2,4)\). The point \(B\) has coordinates \((6,10)\). The point \(C\) has coordinates \((12,2)\).
(a) Find the gradients of the lines \(AB\), \(AC\) and \(BC\).
(b) Hence find the equation of the circle which passes through the points \(A\), \(B\) and \(C\).