Full transcript of “SOLVING REAL-LIFE PROBLEMS USING ARITHMETIC SERIES AND GEOMETRIC SERIES”. Click a timestamp to open the video at that moment. Need another video? Use the YouTube transcript generator.
0:03 [Musika] Hello mga kawat. In this video lesson, we will discuss solving real life problems using arithmetic theories and geometric theories. So first balikan natin yung formula na na-discuss natin on how to find the sum of arithmetic theories. So we will use this formula s sub n is equ n/ 2 multip to the quantity of a sub 1 + a sub n kapag given yung first and last term. At gagamitin naman natin itong formula na to s sub n / 2 * 2 * a sub 1 + the quantity of n - 1 * d. So gagamitin
0:50 natin yan kapag given ang per and the common difference. For geometric theories, gagamitin natin yung formula nito for finite geometric series. If your R is not equal to 1 and your R should be greater than 1, no? So gagamitin natin yung formula na a sub 1 m* 1 - r n / 1 - r. Or pwede rin naman a sub 1* r^ n - 1 / r - 1. So ang r natin diyan is the common range. So this formula is for the infinite no geometric theory. So yun ang gagamitin nating formula. So the real life application of
1:36 arithmetic and geometric series are found in any situation. So kapag merong constant additive change that is the arithmetic theories. The type of theories na gagamitin natin sa problem is arithmetic. At kapag may constant multiplicative change that is geometric theory. So here's the example no'no uh AI generated po 'yung pictures natin kapag kinukuha natin no 'yung sum and then may common difference may additive things. So that is arithmetic theory. So kagaya nil kinompute natin yung kahit total distanceung pagbounce ng ball. So that is geometric.
2:21 Okay. So we will solve the following problem. So let's start with the application of arithmetic series. So this is about saving savings. So Maria, a college student in Manila decided to start a savings plan. He saved Php50 on the first week, Php60 on the second week, PHP 70 on the third week. So and so on. If she continues this pattern how much total money will she have saved after 1 year or that is 52 weeks. So we will interpret first no and get the given. So yung first term natin that is the first week no na-save niya. So yung a sub one natin ay Php50.
3:09 The common difference since yung the save niya for the second week is Php60 minus lang natin yun sa 50. So ang common difference natin ay Php10. So the number of terms o yung hinahanap natin. So the sum of 52 weeks or that is 1 year. So yung end natin ay 52. So yung goal natin is to find the total sum. Since sa given natin ano ang mga nandiyan? So nandiyan yung first term and yung common difference. So yung gagamitin nating formula ay ganyan. No yan 'yung gagamitin nating formula. Kapag given ang first term and the common difference. So ano lang gagawin niyo? Isau-substitute niyo lang 'yung mga given natin, mga given values natin
3:55 sa formula. So s from 52 is equ to yung n papalitan lang natin ng 52 / 2. So that is 52 / 2. So 2 2 * 50. Unahin muna natin dun sa loob ng bracket. So + 52 - 1 * 10. So this is yung n natin is 52. So using your calculator, pwede naman na i-direct sa calculator. Kindly check our answer. So the answer is 15,860. So kapag mano-mano, so unahin niyo muna itong nasa loob ng bracket. So unahin niyo muna to. 51 52 - 1 is 51 * 10. So 510 2 * 50. So that is 100 plus yung 510
4:46 doon and then i-add sila. Tapos 52/ 2 is 26. So 26 times yung sum dito sa loob. So ang sagot natin ay 15,860. So ibig sabihin Maria will save a total of Php15,860 after 1 year. Next it's about tricycle fair increase. So the fair for a tricycle in Caloocan City is Php15 for everying ket an additional PH is charge. If a passenger travel a total distance of 8 km, what is the total pair? So yung per kilometer that is the base payer. So yun yung pinakaumpisa.
5:33 And then kaya sa 8 km so yung kasunod na pito na sucing kilometers kaya sa 8 yung una yung base and then the seven succeeding kilometers. So kung i-interpret natin no yung sa unang kilometer that is Php15 sa pangalawa magdadagdag ng Php3 kaya Php18. Then sa pangatlong k another Php3 ang idadagdag. So 18 + 3 21. So makikita natin na may difference 15 18 21. So ang common difference ay 3. So ano yung mga given natin? So yung first term natin is Php15. The common difference ay Php3. And then pinapahanap sa atin yung total
6:21 sa loob ng ano in what? In a total fair. So it's not 8 weeks in one year. So wait wait for a while. So ko pala nabago na copy-paste doon sa okay so yung n natin is 8. And then so 'yung goal nyo dito is to find the total fair or 'yung S sub 8. Kung gagamitin natin 'yung unang formula 'di ba? 'Yung unang formula dapat given 'yung first and last term. So para ah mabalikan niyo lang 'no yung paano ginagamit yun. So yun ang gagamitin natin sa problem number two natin with the application of arithmetic.
7:08 So again ah hindi pa rin natanggal to. So again ah yung given natin kanina. So yung first term natin ay Php15. The common difference ay Php3. Yung n natinung number of NMs 8. So ang goal natin is to find the total payer. So sabi ko nga, i-try natin gamitin yung unang formula how to find the sum of arithmetic theories. But ang need kasi doon ang kailangan meron siyang last term. So dito sa given natin, wala tayong last term. So kunin natin yung last term. So 'yung pang-last term natin is 'yung 8 pang 8 no 'no? So ibig sabihin so 8 sub 8 'yung kukunin
7:55 natin. 'Yung n natin is 8. So substitute lang natin. So 15 + 8 - 1 3 8 - 1 is 7 * 3 21 + 15 36. So ibig sabihin yung last term natin is 36. So pwede na natin magamit yung unang formula. So substitute lang natin. So yung n natin is 8 / 2. So 15 + 36. What is the answer? Okay. So that is 51. Okay. 4 kasi 8/ 2 is 4. So using your calculator so pwede niyo i-check. Okay. So if we're going to multiply that
8:42 is 204 kasi 51 is 204. So ibig sabihin the total payer for 8 km trip is Php24. Next. So this is now the application of geometric series. So sa social media viral post ito. A social media post about a local bario fiesta goes viral. In the first R, 10 people share it. In the second R, each of those 10 people shares it with three new people. In the third R, each of the people who shared in the second or shared it with three new people. So, and so on. So ganun ang
9:28 scenaryo no. So after an hour yung dami ng tao na nag-share ishe-share ulit nila sa tigtatatlong tao na kakilala nila. So if this trend continues, how many total people have shared the post after 5 hours? So kung ipapakita natin sa unang oras 10 tao no? Sa pangalawa yung 10 tao na yon ang bawat isa sa kanila magse-turn ng tigtatatlong tao. So 10 * 3 30. So pangatlong oras. So yung ah foot no na tao na nag-share is-share din nila sa bawat tatlo. So so 30 is 90.
10:17 O kapag nagtuloy-tuloy yan ang tinatanong diyan is yung after 5 hours. So since yung first term natin is 10 and yung n ang tinatanong diyan is yung 5 and the common ratio is 3 kasi ba 10 is 3 o 90/ 30 is 3. So pwede nating gamitin yung unang formula. So s sub n equ a sub 1* r n - 1 / r- 1. So ito na yung gagamitin natin no. Sub 5 is equ 10. So 3 what is 3 5? User calculator. So 3 5 will be 200.
11:08 Anong sagot? 243. Yes. 243 - 1. So 242. So 10/ 2. So anong sagot? So the answer is 1,210. So after 5 hours ganyan na karami yung makakita nung viral post. So after 6 hours, syempre magta-time free ka ulit diyan. No. So, so sa isang araw ibig sabihin sobrang dami naabot na ng milyon ang nakakita nung viral post. Another uh application for geometric series about growth. So, a ball is
11:53 droppped from a height of 10 ft. So, it bounce. So ibig sabihin na yung initial no na drop niya 10 fet on each bounce it rises to 80% of its previous height. So what is the total vertical distance the ball travels before coming to rest? So yung pictures niyo nakita kanina no parang ganun siya. So the total distance is the initial lap plus the sum of all upward and downward distances travel after the lap. So kukunin natin yung sum of all upward and downward distances. Kasi meron tayong initial drop is yung 10 fet. So idadagdag na lang natin yan sa mao-compute natin doun sa sum of all upward and downward distances. So yan
12:41 yung initial drop. So yung ratio natin dito is 8 80% or 0.8. So ang scenario ba after the first bounce is 10 fet. So first bounce height 10 * 0.8 is common ratio that is 8 fet. So mula sa 8 fet ma-multiply ulit sa 8 6.4 fet and then from 6.4 multip 0.8 and 5.12. So tinuloy-tuloy natin 'yan no. Pag tinuloy-tuloy natin, so makikita natin na yung common ratio natin is less than one. So its bounce includes going up and coming back. So ang ano natin diyan
13:27 formula, imu-multiply natin yung makukuha nating sum sa two. Kasi dalawa yun eh. pag-angat at pagbaba nng bola to times two. So kunin natin gamit kunin natin 'yung sum gamit 'yung formula ng ah infinite. So this infinite kasi 'yung common ratio natin 0.8. So less than 1. So 'yun ang gagamitin nating formula. Ito 'yun. A sub 1 / 1 - r. So 8 / 1-0 1.8. So ito yung makukuha niyo 40. Ito pala yung sum ng bounce height. Ngayon pint pag-angat lang yan eh yung pagbaba. So kaya kailangan natin i-multiply by 2. So ibig sabihin yung sum of all upward
14:13 and downward distances traveled after the drop is 80. Tapos idadagdag natin yung 10 yung initial drop na 10 fet. So ang total niya, total distance niya is 90 fet. Okay? For practice exercises. So the first term of arithmetic series is 4 and the common difference is 5. What is the sum of the first 10 terms? A 255 B 265 C 275 D 285. Okay, the correct answer is 265. Next, find the fifth term of the
15:01 geometric sequence 3 12. Okay. What is the fth term? A 24 B 48 C 48 D 9 the correct answer is 48. Next. Calculate the sum of the first six terms of the geometric series where a sub 1 is equ 5 and r or the common ra is 2. Letter A 155, B 315 C 320 D 630. So the correct answer is letter B 315. Next, what is the sum of the infinite geometric series? 18 + 6 + 2 + 23+
15:52 to the infinity kasi ano to e infinite. So we will use the formula yung palang formula for geometric theory. So letter A 27, B 30, C 36, D 40. So sum ang tinatanong. So the answer is letter A. 27. Last question. A company's profit increases by Php,000 each year. If the profit in the first year was Php500,000, what is the total profit for the company after 5 years? Letter A PH,500,000 Blion. Ch,
16:39 and D4 million. So what is the answer? The correct answer is letter C. Okay. Thank you for watching this video. I hope you learn something. Don't forget to like, subscribe, and hit the bell button para updated kayo for more video tutorial. This is your guide in learning your math lesson, your wow channel. Am
