D. Exercises / Activities ACTIVITY 1Directions: Answer the following word problems involving arithmetic sequence and series. Write your answer inside the box. 1. Find the sum of the first 100 positive odd integers.2. Jenny deposited ₱20,000 on an investment that will give ₱1,750 for every year that the money stays in the account. How much money will she have in her account by the end of year 8?3. Christian decided to start baking chocolate cakes on his spare time during community lockdown because of COVID-19 for his extra income. On his first day, he has five (5) customers who want to avail his chocolate cakes. In continuous rate of orders, his first customer orders 15 chocolate cakes, his second customer orders 18 chocolate cakes, his third customer orders 21 chocolate cakes and so on. If one chocolate cake is being sold at ₱150.00, how much is his total sale for his first day?
Answer (1)
↪ ARITHMETIC SEQUENCE AND SERIES[tex]\small\color{lightblue}\underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:}[/tex]1.) The first 100 positive odd integers are 1,3,5,...,199. For Arithmetic Sequence :Given : a₁ is the first term = 1d is the common difference = 2n is the nth term = 100Solution : an = a₁ + (n-1) da100 = 1 + (100-1) 2 = 1 + (99) 2 = 1 + 198Therefore, the an is = 199.For Arithmetic Series : Given : n is the nth term = 100a₁ is the first term = 1an is the last term = 199Solution :[tex] \small \sf{Sn = \frac{n}{2}( a₁ + an)} \: \: \: \: \: \: \: \: \: \: \: \\ \small \sf{S100 = \frac{100}{2} (1 + 199)} \\ \small \sf{ = 50(200)} \\ \small \sf \color{red}{ = 10000 \: \: \: \: }[/tex]Therefore, the sum of the first 100 positive odd integers is 10,000.[tex]\small\color{lightblue}\underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:}[/tex]2.) How much money will she have in her account by the end of year 8? For Arithmetic Sequence : ₱ 1750, ₱1750, ₱1750, ...a₁ is the first term = 1750d is the common difference = 0 (since the interest is constant each year)For Arithmetic Series :Given : n is the number of terms = 8 (years)a₁ is the first term = ₱1750d is the common difference = 0Solution : [tex]\small \sf{Sn = \frac{n}{2}(2a₁ + n - 1)d } \: \: \: \: \: \: \: \: \: \: \: \: \\ \small \sf{S8 = \frac{8}{2}(2(₱1750) + 8 - 1)0} \\ \small \sf{ = 4(₱3500 + 7)0} \: \: \: \: \: \: \: \: \: \: \\ \small \sf{ = 4(3500)} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \small \sf \color{red}{ = 14000} \: \: \: \: [/tex]Let's calculate the final balanceInitial Deposit + Total Interest= 20,000 + 14,000= 34,000.Therefore, Jenny will have ₱34000 in her account by the end of 8 years. [tex]\small\color{lightblue}\underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:}[/tex]3.) If one chocolate cake is being sold at ₱150.00, how much is his total sale for his first day?15,18,21,...For Arithmetic Sequence :Given : a₁ is the first term = 15d is the common difference = 6n is the nth term = 5 (costumers) Solution : an = a₁ + (n-1) da5 = 15 + (5-1) 6 = 15 + (4) 6 = 15 + (24) = 39.Therefore, the last term is 39.For Arithmetic Series : Given : n is the nth term = 5a₁ is the first term = 15an is the last term = 39Solution : [tex] \small \sf{Sn = \frac{n}{2} (a₁ + an)} \\ \small \sf{S5 = \frac{5}{2} (15 + 39)} \\ \small \sf{ = \frac{5}{2}(54) } \\ \small \sf \color{red}{ = 135} \: \: \: \: \: [/tex]Let's calculate the total sales : Total cakes sold × Price per cake= 135 × ₱150.00= ₱20,250.00[tex]\small\color{lightblue}\underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:}[/tex][tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 07/30/25[/tex][tex]\begin{gathered} \:\begin{gathered}{\begin{gathered} \gamma \\ \huge \boxed{ \ddot \smile}\end{gathered}}\end{gathered} \end{gathered} [/tex]
