Further Maths - Induction for Series
Pearson Edexcel Further Mathematics 2022
Base case and substitutions into the series formula
What do you get if you substitute $k = 1$ for $\frac{1}{2}k(k+1)$?
\[1\]
Writing out and splitting off the last term
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
\[\sum^{k + 1} _ {r = 1} r\]
?
\[1 + 2 + 3 + ... + k + (k + 1)\]
How could you rewrite
\[\sum^{k + 1} _ {r = 1} r\]
?
\[( \sum^{k} _ {r=1} r ) + (k + 1)\]
Using the inductive hypothesis and factorising
How could you rewrite
\[( \sum^{k} _ {r=1} r ) + (k + 1)\]
using the series formula?
\[(\frac{1}{2}k(k+1)) + (k+1)\]
Factorise
\[(\frac{1}{2}k(k+1)) + (k+1)\]
?
\[\frac{1}{2}(k+1)(k+2)\]
Substitute $k = k + 1$ into
\[\frac{1}{2}k(k+1)\]
?
\[\frac{1}{2}(k+1)(k+2)\]