Further Maths - Induction for Series

Created: December 10, 2020

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

\[\sum^{k + 1} _ {r = 1} r\]

?
\[1 + 2 + 3 + ... + k + (k + 1)\]

How could you rewrite

\[\sum^{k + 1} _ {r = 1} r\]

?
\[( \sum^{k}_{r=1} r ) + (k + 1)\]

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

\[\frac{1}{2}k(k+1)\]

?
\[\frac{1}{2}(k+1)(k+2)\]

Simplify

\[k^2(k+1) + (k+1)(3k+2)\]

?
\[(k+1)(k^2 + 3k + 2)\]


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