Arithmetic and Geometric Sequences

Sequence

1. Sequence: A list of numbers arranged in a specific order.
2. Term: Each number in the sequence is called a term of the sequence. These are named as the first term, second term, third term, , -th term, , or written as the st term, nd term, rd term, , -th term, .
3. General Term: A sequence is typically written as , and the -th term is called the general term of the sequence. A sequence with the general term is written as .

Arithmetic Sequence

1. Arithmetic Sequence: A sequence created by successively adding a constant value to the preceding term.
2. Common Difference: The constant value added in an arithmetic sequence.
3. General Term of an Arithmetic Sequence: If the first term is and the common difference is , the general term of the arithmetic sequence is:
   
4. Arithmetic Mean: If three numbers , , and form an arithmetic sequence in this order, is called the arithmetic mean of and . Since , we have:
   
5. Sum of an Arithmetic Sequence
1. The sum of the first terms of an arithmetic sequence is:
1. If the first term is and the -th term is , then:
   
2. If the first term is and the common difference is , then:
   
2. Relationship between the sum and the general term of a sequence:
   For the sequence , the sum of the first terms is given by:
   
   If (where are constants), then:
1. If , the sequence forms an arithmetic sequence from the first term.
2. If , the sequence forms an arithmetic sequence from the second term.

Geometric Sequence

1. Geometric Sequence: A sequence created by successively multiplying each term by a constant value.
2. Common Ratio: The constant value multiplied in a geometric sequence.
3. General Term of a Geometric Sequence: If the first term is and the common ratio is (where ), the general term of the geometric sequence is:
   
4. Geometric Mean: If three non-zero numbers , , and form a geometric sequence in this order, is called the geometric mean of and . Since , we have:
   
5. Sum of a Geometric Sequence
1. The sum of the first terms of a geometric sequence with first term and common ratio is:
1. If , then:
   
2. If , then:
   
2. Relationship between the sum and the general term of a sequence:
   For the sequence , the sum of the first terms is:
   
   
   If (where , , and are constants), then:
1. If , the sequence forms a geometric sequence from the first term.
2. If , the sequence forms a geometric sequence from the second term.
3. Interest Calculation: If an amount is deposited for years at an annual interest rate , the total amount is:
1. Simple Interest:
2. Compound Interest: