Arithmetic Sequences and Polynomial Division

Saving Plans and Sequences

• Reign's saving method involves adding 5 pesos daily, starting with 10 pesos, categorizing it as an Arithmetic Sequence with a common difference of 5.
• Arithmetic sequences are characterized by a constant difference between consecutive terms.

Examples of Sequences

• Classic examples of Arithmetic Sequences include:
  • 5, 10, 15, 20... (constant difference of 5)
• Geometric Sequences are identified by a constant ratio, such as:
  • 3, 6, 12... where each term is multiplied by 2.

Identifying Sequences

• The sequence 6, ___, ___, ___, 22 can be figured out, with options including:
  • 10, 13, 16 as a valid progression with a common difference of 3.
• To find the 5th term of the sequence expressed as ( a_n = 3n + 8 ), substituting ( n = 5 ) yields 23.

Polynomial Division

• Synthetic division is a method used to divide polynomials and to find specific quotients.
• The remainder theorem states that the remainder of a polynomial ( P(x) ) divided by ( (x - c) ) is equal to ( P(c) ).

Specific Calculations

• Liam follows a pattern of buying marbles: 4, 8, 12, 16... leading to him collecting 48 marbles on day 12.
• Identifying the largest common factor for ( 12^2 ) and ( 8^5 ) leads to ( 4^2 ).

Geometric Mean and Quotients

• The geometric mean of -3 and -108 is calculated as -18.
• When dividing ( (x^2 + x - 2) ) by ( (x - 1) ), the result is ( x + 2 ).

Area and Factorization

• The area of a rectangle with length ( (3x - 7) ) cm and width ( (4x + 5) ) cm can be derived from the product of the two expressions.
• Complete factorization of ( 2^d - 32 ) results in ( 4 ).

Real-Life Applications

• A house purchased for Php 800,000 in 2006 appreciates by Php 15,000 yearly, reaching Php 1,040,000 by the end of 2022.
• Mr. Santos saves Php 2,500 monthly on top of an existing Php 15,000, accumulating amounts of Php 17,500, Php 20,000, Php 22,500, and Php 25,000 after successive months.

Egg Collection Problem

• A farmer uses eight baskets with an exponential increase in the number of eggs: three in the first, six in the second, and twelve in the third, doubling in subsequent baskets, culminating in a total collection of 765 eggs.

This quiz focuses on understanding arithmetic sequences, their characteristics, and examples. Additionally, it covers polynomial division and the remainder theorem, providing a comprehensive assessment of these mathematical concepts.