Application of Algebra

Collaborative Learning: Learners to work in convenient groups (ability, mixed-ability, mixed gender, etc.) to solve problems and present answers
Talk for Learning Approaches: Learners brainstorm and discuss the types and features of sequences.
Project-based Learning: Engage learners to come up with a sequence that is relevant and applicable to real life. Experiential Learning: Engage learners with hands-on activity (learning by doing).
Problem-based learning: Learners inquire into what happens when terms of sequences are summed up to infinity.
Activity 1: Facts about Geometric and Arithmetic sequences.
Initiate Talk for Learning Approaches (building on what others say, managing Talk for Learning, Structuring Talk for Learning); Experiential and Collaborative learning approaches to support revised facts about arithmetic and geometric sequences. Starting with whole group discussion and transiting through small convenient groups (e.g., mixedability/mixed gender) or stations/centres, then pair work (e.g., using think-pair-share) and finally individual (e.g., using give-one take-one) learners recall, assess and debate each other on what they know about sequences including how to find the nth term, means of sequences and solve problems related to sequence and series in different contexts.
Example Learners discuss, debate, assess and challenge colleagues on: i) examples of geometric and arithmetic sequences (both mechanical and real life examples)
ii) how arithmetic and geometric sequences are similar; iii) how arithmetic and geometric sequences are different; iv) examples of sequences that are neither geometric nor arithmetic and explain how to solve such sequences.
! ! ! The first four terms of a geometric sequence are given: − ") , 1 , − ) , 2 ... what is the 5 th term in the
sequence?
Activity 2: Sums in a Sequence
Use Collaborative Learning, Experiential Learning, Problem-based Learning and Talk for Learning Approaches: Learners work in convenient groups to study, interpret and use summation notation/symbols to generate terms in a sequence.
Example: Indicate the first three terms in the series ∑ 7%4! (4𝑛 − 5) .
Solution: To find the first three terms, replace 𝑛 with 1, 2 and 3: ∴ The first three terms are -1, 3 and 7 ∑ !(
- %4! (4𝑛 − 5) .
- ∑ 2l4*) (2) l . !
- ∑ !! &4) ) (4) &*) .
Activity 3: Method of undetermined coefficient
Use Collaborative Learning, Experiential Learning, Problem-based Learning and Talk for Learning Approaches: Learners use the Method of Undetermined Coefficient to find the sum of the first 𝑛 terms of other series by equating identically the given series to a series of the form 𝐴 + 𝐵 + 𝐶𝑛 ) + ... ... ..., and then determining the values of the constants 𝐴, 𝐵, 𝐶
Example Find the sum of the first n natural numbers.
Solution: 𝑛 = 1, 2, 3, 4, ... ... ... ... ∞ The sum of the first n natural number can be written as:
∑ %34! 𝑟 = 1 + 2 + 3 + ⋯ + 𝑛 = 𝐴 + 𝐵 + 𝐶𝑛 ) ...(1) Adding (𝑛 + 1) natural numbers gives: ∑ %34! 𝑟 = 1 + 2 + 3 + ⋯ + 𝑛 + (𝑛 + 1) = 𝐴 + 𝐵 + 1 + 𝐶(𝑛 + 1) ) ...(2) %(%$!) Solving equation (2)-(1) and comparing coefficients we have ∑ %34! 𝑟 = 1 + 2 + 3 + ⋯ + 𝑛 = )
)
- Evaluate ∑ %34! ()3$!)()3*!) ;
Hint: Split into partial fractions and solve.
Activity 4: Sum of AP and GP
Use Collaborative Learning, Experiential Learning, Problem-based Learning and Talk for Learning Approaches: % Learners work collaboratively to investigate, establish, and use the formula 𝑆 = ) (𝑎 ! + 𝑎 %) ) and 𝑆 % = ) [(2𝑎 ! + (𝑛 − 1)𝑑] to find the sum of an AP and solve related problems; where 𝑛 is the number
of terms in the series, 𝑎 ! is the first term and 𝑎 % is the last term, and d is the common difference of an AP.
Example Find the sum of all even integers from 250 to 350.
Solution: 𝑎 ! = 250; 𝑎 % = 350, 𝑛 = 50 .( 𝑆 .( = ) (250 + 350) = 15,000
A supermarket displayed milk tins piled up in the form of a pyramid. The bottom layer has 25 tins, and each successive layer has 2 fewer tins. How many milk tins are displayed?
& & $ (3 # *!) Learners work collaboratively to investigate, establish, and use the formula 𝑆 c = !*3 $ ; 𝑆 % = ; 𝑟 > 3*! & $ (!*3 # ) 1 or 𝑆 % = ; 𝑟 < 1 to find the sum of a GP and solve related problems, where 𝑛 is the !*3 number of terms in the series, 𝑎 ! is the first term and r is constant ratio.
Example ) , 1 Evaluate: 1 − " + 2 − )7 + ...
Indicate the sum of the following series: i) ∑ 2l4*) (2) l .
ii) ∑ !! ! #*) #4) i ) (4) j .
Activity 5: Convergence or divergence of a series:
Use Collaborative Learning, Experiential Learning, Problem-based Learning and Talk for Learning Approaches: Learners work in convenient groups, investigate whether a series converges, and justify their claims.
Example: ! ! ! ! ! Consider the series ∑ c 34! ) 0 = ) + , + 1 + ⋯ + ) # + ⋯
Adding the terms in the series gives: ! ! ! " ! ! ! 7 !. S 1 = ) ; S 2 = ) + , = , ; S 3 = ) + , + 1 = 1 ; S 4 = !/ ; "! /" !)7 S 5 = ") ; S 6 = /, ; S 7 = !)1 ; ... and so on.
It is obvious the sum of the terms in the series (𝑆 % ), always less than 1 but gets closer to 1 as we take more and more terms. It is reasonable to claim that ∑ c ! ! ! ! ! = + + + ⋯ + # + ⋯= 1 34! ) 0 ) , 1 )
$ ! ! Note that the above series is a G. P. with a = and common ratio r = < 1. Verify that S ∞ = ! =1 ) ) !* $ !
Note: S 1, S 2, S 3, S 4 ...are the partial sums of the series. If there is a number L such that S n = ∑ c m4! S ] = L. The number L is called the sum of the infinite series. If there is no such number, then the series is said to diverge.
Find the sum, if possible: 1−3+9−27+...
Teaching and Learning Resources:
- Worksheets
- Scientific Calculator
- Technological tools, apps, etc.
- SHS Additional Mathematics Curriculum
Assessment (2.1.1.AS.1). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 2 Skills of conceptual understanding; Level 3 Strategic reasoning; Level 4 Extended critical thinking and reasoning.