Arithmetic Progressions
An Arithmetic Progression (AP) is a sequence of numbers where the difference of any two successive terms is a constant $d$, called the common difference.
An AP can be denoted as $a_1=a$ and $a_n=a_{n-1}+d$, where $a_n$ is the n'th term of the progression.
For example, in the progression 1, 3, 5, 7, 9, ..., the first term is $a_1=1$, and the n'th term is given by $a_n=a_{n-1}+2$.
Sum of the "n" first terms of an AP
Let's call $S$ the sum of the "n" first terms of an AP with common difference $d$. Thus:
$S=a_1+a_2+...++a_{n-1}+a_n$. This sum can also be writen as:
$S=a_n+a_{n-1}+...+a_2+a_1$
Adding the two equations:
$2S=(a_1+a_n)+(a_2+a_{n-1})+(a_3+a_{n-2})+...+(a_n+a_1)$
$2S=(a_1+a_n)+(a_1+a_n)+(a_1+a_n)+...+(a_1+a_n)$
$2S=n(a_1+a_n)$
$S=\frac{n(a_1+a_n)}{2}$
Note that,
$a_2+a_{n-1}=a_1+d+a_n-d=a_1+a_n$
$a_3+a_{n-2}=a_1+2d+a_n-2d=a_1+a_n$
and so on.
Geometric Progressions
A Geometric Progression (GP) is a sequence of numbers such that each term after the first equals the previous term multiplied by a constant "q", called the common ratio.
A GP can be denoted as $a_1=a$ and $a_n=qa_{n-1}$, where $a_n$ is the n'th term of the progression.
For example, in the progression 2, 6, 18, ..., the first term is $a_1=2$, and the n'th term is given by $a_n=a_{n-1}3$.
Sum of the "n" first terms of a GP
Let's call $S$ the sum of the "n" first terms of a GP with common ratio "q". Thus:
$S=a_1+a_2+a_3+...+a_n$
$S=a_1+qa_1+q^2a_1+...+q^{n-1}a_1$
Multiplying both sides by $q$:
$Sq=qa_1+q^2a_1+q^3a_1+...+q^na_1$
This equation is equal to the previous equation without the first term, $a_1$, and with the addition of the new term $q^na_1$. Therefore:
$Sq=S-a_1+q^na_1$
$S(q-1)=(q^n-1)a_1$
$S=a_1\frac{q^n-1}{q-1}$
Product of the "n" first terms of a GP
Let's call $P$ the product of the "n" first terms of a GP with common ratio "q". Thus:
$P=a_1a_2a_3...a_n$
$P=a_1(qa_1)(q^2a_1)...(q^{n-1}a_1)$
$a_1$ repeats itself "n" times in the equation. Thus:
$P=a_1^n(qq^2...q^{n-1})$
$P=a_1^nq^{1+2+...+(n-1)}$
The sum of the terms of an Arithmetic Progression is given by $S=\frac{n(a_1+a_n)}{2}$. Thus:
$Exponent=1+2+...+(n-1)$
$Exponent=\frac{(n-1)(1+n-1)}{2}$
$Exponent=\frac{(n-1)n}{2}$
Plugging this result into the previous equation:
$P=a_1^nq^{1+2+...+(n-1)}$
$P=a_1^nq^{\frac{(n-1)n}{2}}$
Solved SAT Practice Tests
Find Practice Tests in the following link:
SAT Practice Tests - Arithmetic and Geometric Progressions and
Additional Practice Tests - Arithmetic and Geometric Progressions
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