@?induction

What is the general technique for proving a statement about matricies using induction?? Multiplying out the matrix for $k+1$ and showing that it’s the same as substituting $k = k+1$ into the normal $k$ matrix. How could you rewrite $M^{k+1}$?? $$ M^k M $$ Backlinks [[Further Maths - Syllabus]]S Metadata date: 2021-01-05 12:26 tags: - '@?further-maths' - '@?induction' - '@?public' title: Further Maths - Induction for Matricies

If you add a multiple of $4$ to something already divisible by $4$, what must be true about the answer?? It is also divisible by 4. What is the two-step general technique used to show divisibility in induction?? Assume $f(k)$ is divisible Show the difference between $f(k)$ and $f(k+1)$ is divisible. How would you write the difference between $f(k)$ and $f(k+1)$?? $$ f(k+1) - f(k) $$ If $$f(k + 1) - f(k) = 4\times 3^{2k}$$, what other statement could you write that shows clearly $f(k+1)$ is divisible by $4$?

What is $$\sum^{1}_{r = 1} r$$?? $$ 1 $$ What do you get if you substitute $k = 1$ for $\frac{1}{2}k{k+1}$?? $$ 1 $$ How could you write out the sum that is being done for $$\sum^{k}_{r = 1} r$$?? $$ 1 + 2 + 3 + … + (k - 1) + k $$ How could you write out the sum that is being done for $$\sum^{k + 1}_{r = 1} r$$?? $$ 1 + 2 + 3 + … + k + (k + 1) $$

See also: [[Further Maths - Induction for Series]]S [[Further Maths - Induction for Divisibility]]S What is induction?? A proof technique that shows a statement is true for natural numbers. How could you use induction to prove you can climb as high as you like on a ladder?? You can climb onto the bottom rung You can climb onto the next rung What are the 4 steps for induction?