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Basis : The basis of a vector space is a set of linearly independent vectors, so that every vector of the vector space can be written as a linear combination of the vectors in the basis. For example $\{\left(\begin{smallmatrix}1\\0\\0\\\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\1\\0\\\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\0\\1\\\end{smallmatrix}\right)\}$ is a basis of $\mathbb{R}^3.$ In a prove by induction basis refers to the first step in which you have to prove that the result is true for a particular number $n$ (usually $n=0$ or $n=1$.)
Equation : An equation is a mathematical statement in which two expressions are written with an equal sign in between. A solution of an equation is a set of variables that makes the statement a true statement.
Half : A Half is the number equal to $\frac{1}{2}=0.5.$
Induction : Induction is a common techniques for proofing mathematical results that are dependent on a natural number $n.$ It consists of first proving the result for a specific $n$, often $n=0$ or $n=1$ (the induction base) and then proving that if it the results holds for all numbers smaller than $n$ that it will then also hold true for $n$ (the induction step.)
Proof : A proof is a logical deduction of a result starting only with the assumptions of the result.
Quarter : One quarter is the number $\frac{1}{4}=0.25.$
Third : A third either refers to the third object in an ordering or to the number $\frac{1}{3}=0.\overline{3}.$
We want to proof by induction that $\frac{1}{2}+\frac{1}{4}+\ldots+\frac{1}{2^n}=\frac{2^n-1}{2^n}$ for all $n\geq 1.$ What could be the basis for the induction?
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