The Quadratic Formula

Introduction

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Linear and Quadratic Inequalities

Introduction

 

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Geometric Progression, Series & Sums

Introduction

A geometric sequence is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r. The common ratio (r) is obtained by dividing any term by the preceding term, i.e.,

where	r	common ratio
	a1	first term
	a2	second term
	a3	third term
	an-1	the term before the n th term
	an	the n th term

The geometric sequence is sometimes called the geometric progression or GP, for short.

For example, the sequence 1, 3, 9, 27, 81 is a geometric sequence. Note that after the first term, the next term is obtained by multiplying the preceding element by 3.

The geometric sequence has its sequence formation:

To find the nth term of a geometric sequence we use the formula:

where	r	common ratio
	a1	first term
	an-1	the term before the n th term
	n	number of terms

Sum of Terms in a Geometric Progression

Finding the sum of terms in a geometric progression is easily obtained by applying the formulas:

nth partial sum of a geometric sequence

sum to infinity

where	Sn	sum of GP with n terms
	S∞	sum of GP with infinitely many terms
	a1	the first term
	r	common ratio
	n	number of terms

Examples of Common Problems to Solve

Write down a specific term in a Geometric Progression

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Write down the 8th term in the Geometric Progression 1, 3, 9, ...

Answer

Finding the number of terms in a Geometric Progression

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Find the number of terms in the geometric progression 6, 12, 24, ..., 1536

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Finding the sum of a Geometric Series

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Find the sum of each of the geometric series

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Finding the sum of a Geometric Series to Infinity

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Converting a Recurring Decimal to a Fraction

Decimals that occurs in repetition infinitely or are repeated in period are called recurring decimals.

For example, 0.22222222... is a recurring decimal because the number 2 is repeated infinitely.

The recurring decimal 0.22222222... can be written as .

Another example is 0.234523452345... is a recurring decimal because the number 2345 is repeated periodically.

Thus, it can be written as or it can also be expressed in fractions.

Question

Express as a fraction in their lowest terms.

Answer

 

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Indices & the Law of Indices

Introduction

Indices are a useful way of more simply expressing large numbers. They also present us with many useful properties for manipulating them using what are called the Law of Indices.

What are Indices?

The expression 2⁵ is defined as follows:

We call "2" the base and "5" the index.

Law of Indices

To manipulate expressions, we can consider using the Law of Indices. These laws only apply to expressions with the same base, for example, 3⁴ and 3² can be manipulated using the Law of Indices, but we cannot use the Law of Indices to manipulate the expressions 3⁵ and 5⁷ as their base differs (their bases are 3 and 5, respectively).

Six rules of the Law of Indices

Rule 1:

Any number, except 0, whose index is 0 is always equal to 1, regardless of the value of the base.

An Example:

Simplify 2⁰:

Rule 2:

An Example:

Simplify 2^-2:

Rule 3:

To multiply expressions with the same base, the base and add the indices.

An Example:

Simplify : (note: 5 = 5¹)

Rule 4:

To divide expressions with the same base, the base and subtract the indices.

An Example:

Simplify

:

Rule 5:

To raise an expression to the nth index, the base and multiply the indices.

An Example:

Simplify (y²)⁶:

Rule 6:

An Example:

Simplify 125^2/3: