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How could you imagine any function $f(x)$ could be written as a polynomial?? $$ f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + … + a_r x^r $$ $f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + … + a_r x^r$ If you wanted to work out $a_0$, what could you set $x$ equal to?? $$ 0 $$ $f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + … + a_r x^r$ What happens if you substitute in $x = 0$?

See Also [[Further Maths - Series]]S Flashcards What is the requirement of a series for the method of differences to be applicable?? The general term, $u_r$ of a series can be expressed in the form $f(r) - f(r+1)$. If the general term of a series $u_r$ can be expressed as $f(r) - f(r + 1)$, how could you write the series?? $$ \sum^n_{r = 1} (f(r) - f(r + 1)) $$

What is the formula for the sum of the cubes of the first $n$ natural numbers?? $$ \sum^{n}_{r=1} r^3 = \frac{1}{4}n^2(n+1)^2 $$ How could you rewrite $\sum^{n}_{r=1} r^3$?? $$ \frac{1}{4}n^2(n+1)^2 $$ What’s another way of expressing $\frac{1}{4}n^2(n+1)^2$?? $$ \sum^{n}_{r=1} r^3 $$ How could you rewrite $\sum^{n}_{r=1} 4r^2$?? $$ n^2(n+1)^2 $$ What’s an easy way for remembering the sum of cubes formula?? It’s the sum of the natural numbers formula squared. Backlinks [[Further Maths - Series]]S [[Further Maths - Syllabus]]S Metadata date: 2020-09-16 17:57 tags: - '@?

What is the formula for the sum of the squares of the $n$ natural numbers?? $$ \sum^{n}_{r=1} r^2 = \frac{1}{6}n(n+1)(2n+1) $$ How could you rewrite $\sum^{n}_{r=1} r^2$?? $$ \frac{1}{6}n(n+1)(2n+1) $$ What’s another way of expressing $\frac{1}{6}n(n+1)(2n+1)$?? $$ \sum^{n}_{r=1} r^2 $$ How could you rewrite $\sum^{n}_{r=1} 3r^2$?? $$ \frac{1}{3}n(n+1)(2n+1) $$ Backlinks [[Further Maths - Series]]S [[Further Maths - Syllabus]]S Metadata date: 2020-09-16 17:57 tags: - '@?further-maths' - '@?series' - '@?

Sums of Natural Numbers What are the natural numbers?? $1, 2, 3, 4, 5, 6\ldots$ What are the whole numbers?? $0, 1, 2, 3, 4, 5, 6\ldots$ What’s another name for the sum of the first $n$ natural numbers?? The triangle numbers. How can you visualise the sum of the first $n$ natural numbers?? As a triangle. How can you visualise the sum of the first $n$ natural numbers, doubled?? As a $n \times (n+1)$ square.

Series Core 1 [[Further Maths - Sums of Natural Numbers]]S [[Further Maths - Sums of Squares]]S [[Further Maths - Sums of Cubes]]S [[Further Maths - Series Tips and Tricks]]S [[Further Maths - Induction for Series]]S Core 2 [[Further Maths - The Method of Differences]]S Flashcards What is a series?? A sum of sequential terms. What is the notation for series?