WORD PROBLEMS about finding the nth term of a geometric sequence: 1. Doubling Coins Carla has 5 peso coins on the first day. Every day, she doubles the number of coins. How many coins does she have on the 6th day? 2. Halving Chocolate A chocolate bar weighs 80 grams. Each day, Jerry eats half of what’s left. How much chocolate is left on the 4th day? 3. Tripling Money Marco invests Php ₱1,000. Every year, his money triples. How much money does he have after 5 years? WORD problems involving Geometric Sum. 1. Finite Geometric Sum (Simple Doubling) Anna saves ₱2 on the first day. Each day, she doubles her savings. How much has she saved after 6 days? 2. Finite Geometric Sum (r = 1 Case) Ben earns Php 5,000 each week for 6 weeks. How much does he earn in total? (This is a geometric sum with r = 1.) 3. Finite Geometric Sum (r = -1 Case) A light flashes on and off alternately. If the pattern is ON, OFF, ON, OFF, ON, and each flash counts as 1, what is the total sum after 5 flashes, using 1 for ON and -1 for OFF? (This uses r = -1.) 4. Infinite Geometric Sum (Simple Fraction) A small ball bounces back to half of its previous height each time. If it’s dropped from 8 meters, what is the total distance if it will bounce in infinite bounces? (Infinite geometric sum, r = 1/2)​

Answer:Step-by-step explanation:Formula for geometric sequence:[tex]$$ a_n = a_1 \cdot r^{(n-1)} $$[/tex]Formula for finite geometric sum:[tex]$$ S_n = a_1 \cdot \frac{r^n - 1}{r - 1} $$[/tex]Formula for infinite geometric sum:[tex]s=\frac{a_1}{1-r}[/tex]1. 5coins, double each day. Coin on 6th day[tex]$$ a_n = a_1 \cdot r^{(n-1)} $$[/tex][tex]a_6[/tex] = 5(2^5)=5(32)=160Therefore, she has 160 coins on the 6th day2. 80 grams, halving each day. Grams on the 4th day[tex]$$ a_n = a_1 \cdot r^{(n-1)} $$[/tex][tex]a_4[/tex] = 80(1/2^3)= 80(1/8)= 10Therefore, he'd have 10 grams of chocolate left on day 4.3. Money triples, invest 1000 each year. How much in 5 years?                [tex]$$ a_n = a_1 \cdot r^{(n-1)} $$[/tex][tex]a_5[/tex] =  1000(3^5-1)=1000(81)=81000Therefore, he has 81,000 pesos in 5 yearsGeometric sumsFinite1. P2 each day, doubles, how much after 6 days[tex]$$ S_n = a_1 \cdot \frac{r^n - 1}{r - 1} $$[/tex][tex]s_6[/tex]= (2) [tex]\frac{2^{6} -1}{2-1}[/tex]=(2) 63/1=2(63)=126Therefore, She saved P126 after 6 days.2.5,000 each week, for 6 weeks. Earn in total?if r = 1, just multiply the nth term with the first term5,000(6)=30,000Therefore, He earns 30,000 in 6 weeks3. on,off,on,off,on. on =1, off =-1. Sum after 5 flashess = 1+-1+1+-1+1= 1Therefore, the sum after 5 flashes is 14. Infinite, ball bounce back half its previous height. Dropped from 8 meters, total distance in infinite bounces.[tex]s=\frac{a_1}{1-r}[/tex]= [tex]\frac{8}{1-\frac{1}{2} }[/tex]=[tex]\frac{8}{\frac{1}{2} }[/tex]= 16Therefore, the total distance of the ball if it bounces in infinite bounces is 16 meters

Answered by keinasour | 2025-07-20