Mathematics Grade 10: Illustrating an Arithmetic Sequence

Arithmetic Sequence: A sequence where each term after the first is obtained by adding a fixed constant number to the preceding term.
Common Difference ( $d$ ): The fixed constant added. It is calculated by subtracting a term from the following term ( $d = a_n - a_{n-1}$ ).
Verification: A sequence is arithmetic if ( $a_2 - a_1 = a_3 - a_2 = a_4 - a_3$ ).

Finding the Common Difference ( $d$ ): * For the sequence $-4, 3, 10, 17, \dots$ , $d = 3 - (-4) = 7$ . * For the sequence $3a - 1, 3a, 3a + 1, \dots$ , $d = 3a - (3a - 1) = 1$ .
Predicting Terms: For the sequence $5, 3, 1, -1, -3, \dots$ , the first term ( $a_1$ ) is $5$ and $d = -2$ .
Solving for Variables: To make $3a + 1, 4a, 6a + 1$ an arithmetic sequence: * $4a - (3a + 1) = (6a + 1) - 4a$ * $a - 1 = 2a + 1$ * $a = -2$
Calculating $d$ from specific terms: If $a_1 = 1$ and $a_4 = 7$ : * $a_4 = a_1 + 3d$ * $7 = 1 + 3d \rightarrow d = 2$

PUJ Modernized Fare (LTFRB, April 24, 2020): * Distance: $1^{st}$ km = $11.00$ ; $2^{nd}$ km = $12.50$ ; $3^{rd}$ km = $14.00$ ; $4^{th}$ km = $15.50$ . * This is an arithmetic sequence with $d = 1.50$ .
Walking Exercise Pattern: * A person walks $40\,m$ on day 1, $60\,m$ on day 2, and $80\,m$ on day 3. * Following this pattern ( $d = 20\,m$ ), the distance on the $10^{th}$ day is $220\,m$ .

Facilitator and Learner Guidance: This module (M10AL - Ib – 1) is designed to help learners meet K to 12 Curriculum standards by defining and illustrating arithmetic sequences.
Development Team: * Writer: Evelyn N. Ballatong * Editor: Heather G. Banagui * Reviewer: Bryan A. Hidalgo * Management Team: May B. Eclar, PhD; Benedicta B. Gamatero; Carmel F. Meris; Marciana M. Aydinan, PhD; Ethielyn E. Taqued; Edgar H. Madlaing; Lydia I. Belingon.
Legal: Published by the Department of Education, Secretary Leonor Magtolis Briones and Undersecretary Diosdado M. San Antonio. Subject to Republic Act 8293, section 176.