# Further Maths - Induction for Series

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/induction-for-series/ · Updated: 2020-12-10 · Tags: further-maths, induction, year-1

##### 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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