Observe that 1 = 1, 1 − 4 = −(1 + 2), 1 − 4 + 9 = 1 + 2 + 3, 1 − 4 + 9 − 16 = −(1 + 2 + 3 + 4), 1 − 4 + 9 − 16 + 25 = 1 + 2 + 3 + 4 + 5. Guess a general formula and prove it by mathematical induction.
Percentage
A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.
Algebraic Expressions
In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.
Numbers
Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.
Subtraction
Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.
Addition
Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.
Observe that
1 = 1,
1 − 4 = −(1 + 2),
1 − 4 + 9 = 1 + 2 + 3,
1 − 4 + 9 − 16 = −(1 + 2 + 3 + 4), 1 − 4 + 9 − 16 + 25 = 1 + 2 + 3 + 4 + 5.
Guess a general formula and prove it by mathematical induction.
2) a) Find a formula for the sum of the first n of nonnegative integers that are multiple of 10.
b) Prove the formula that you conjectured in part (a).
3) For any non negative integer n , prove that an − bn is divisible by a−b,where a and b are any integers with a not equal b.
4) Prove that 2n + 3 ≤ 2n if n is an integer greater than 3.
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