Application of Sum of an AP | Sum of Arithmetic Progression Formula Mathematics Tutorial Video

Added: 2026-05-08

[00:00:00]Let's look at this question on the sum of an AP or rather the application of the sum of an AP. The question here says, "Two men, A and B, started work at the same time.

[00:00:14]A at a salary of 400,000 Naira and an with an annual increment of 5,000 Naira and also B with an with at a salary of 500,000 Naira with an annual increment of 4,000 Naira.

[00:00:36]After how many years will their total salaries and be equal?" All right.

[00:00:43]All right, let's see how that works.

[00:00:46]The solution now.

[00:00:50]Solution.

[00:00:53]So, you're having man A and man B.

[00:00:56]Now, for man A let's start with for man A. For for man A the salary he earns, A, is equal to 400,000 Naira.

[00:01:09]400,000 Naira and the annual increment, that's the difference, D, is equal to 5,000 Naira.

[00:01:21]That's the value.

[00:01:23]For B, for man B, that's for B his salary, A, is about 500,000 Naira.

[00:01:31]That's 500,000 Naira and his annual increment, the difference in his annual increment, is just 4,000 Naira.

[00:01:42]All right, I think that was what they said here.

[00:01:45]Let's see.

[00:01:48]Yeah, [snorts] that's what they said here.

[00:01:49]And they're asking, "After how long will their will will what? Will their total salary be Will their Will their total salaries and be equal?" That's the question. "After how many years will their total salaries and be equal?" All right, so let's solve this question.

[00:02:09]Okay, so how do we solve this question?

[00:02:29]Now, to solve this question, let's let's get the sum, right, of the salary for A.

[00:02:35]Right, sum of salary for A. Sum of salary for man A.

[00:02:46]We have SN. We said the sum of an for of an AP Let me write the formula here. Sum of an AP, SN, is equal to n over 2 into um 2A into 2A plus n minus 1 d. We'll have this.

[00:03:07]All right, so for A we'll have Let's call this S of A.

[00:03:12]S of A will be equal to um n over 2 or let's call it S of n for A, better still. S of n, so that you don't get confused. So, S of n for A is equal to the same n over 2 into 2A.

[00:03:27]For man A, A is 400,000. So, into 2A 2 into 400,000 All right, plus n minus 1 plus n minus 1 into d.

[00:03:43]>> [snorts] >> Okay, let's get d for man A. D for man A is 5,000.

[00:03:49]So, this into 5,000. So, we have this.

[00:03:54]>> [snorts] >> All right, can we break this down?

[00:03:56]This will be equal to n over 2 into This will give you 2 * 400,000 is 800,000 thousand plus 5,000 * n is 5 thousand n minus 5,000 * 1 is 5 thousand. So, we have this.

[00:04:23]So, this is equal to n over 2 into 8 800,000 minus 500 minus 5,000. That will give you 795,000.

[00:04:37]This then plus 5,000 n. You have this. All right.

[00:04:47]Yep, 800,000 minus 5,000 gives you 795,000 then plus bring this down. So, this is for A. This is the sum of the salary for A. Let's get for B.

[00:05:02]Yep, let's get for B. So, sum sum of salary for B.

[00:05:14]For B, what do you have there? We have SN of B is equal to n over 2 n over 2 into 2 * A 2 * For B, for man B A is 500,000.

[00:05:32]Okay, so 2 * 500,000 500 1 2 3 500,000 plus n minus 1 into d.

[00:05:45]For B, d is 4,000.

[00:05:48]So, into 4,000.

[00:05:52]All right, so we have this.

[00:05:54]Let's expand further.

[00:05:57]If we expand further we have um SN the sum of the salary for B has [clears throat] been equal to n over 2 into 2 * 500,000 gives you 1 million.

[00:06:16]That's 1 million in Naira.

[00:06:20]1 million Naira.

[00:06:22]plus n * 4,000 gives you 4,000 n minus 1 * 4,000 is 4 thousand. So, we have this.

[00:06:41]All right.

[00:06:42]So, this will be n the sum of the salary for B will be n over 2 into 1 million minus 4,000 gives you 996,000.

[00:07:02]Yeah?

[00:07:04]Yes, plus 4 thousand n. So, we have this. All right.

[00:07:14]So, we have this.

[00:07:17]All right, guys, so what next? What next? What next? So, at this juncture um next we do they said for them to be equal.

[00:07:24]um For their salaries for their salaries to be equal to be equal what will happen is that the sum of the salary for A the sum of the salary for A will be equal to the sum or total Yep, total salary for man B. That's it.

[00:07:54]So, let's equate them. Let me get a chalk.

[00:07:58]Let's equate them.

[00:08:03]So, I also [snorts] have my calculator here. For A this value is this.

[00:08:08]Right, um yep, we have um n over 2 into 795,000 plus 5,000 n. This is equal to for B we have this.

[00:08:29]n over 2 into this one here.

[00:08:34]996,000 thousand plus 4,000 n. So, we have this.

[00:08:44]All right.

[00:08:46]Again, we have this and this. Whenever you have the same thing, n over 2 n over 2 on the same on opposite side of an equation at the same position, all right, that means this one here is multiplying this and this one here is multiplying this. So, basically they're the same thing. This can cancel this. All right, the same term at the same position on both side of an equation, they can cancel out.

[00:09:12]With this, now I am left with 795,000.

[00:09:18]This plus 5,000 n is equal to 996,000 plus 4,000 n.

[00:09:34]At this juncture, we say collect like terms. So, collect like terms.

[00:09:47]Let's collect like terms. What do we have? Um Take this one off.

[00:10:01]>> If we take like terms, what we what do we have in there?

[00:10:03]>> [snorts] >> Um I'll move the terms having n here becomes 5,000 n.

[00:10:09]So 5,000 n You have 4,000 on the other part here.

[00:10:17]Move the 4,000 over here becomes negative. So minus 4,000 n is equal to Here you have 996,000.

[00:10:30]So becomes 996,000.

[00:10:33]996 1 2 3 minus This man is positive.

[00:10:40]Move it over here becomes negative. So minus 795 1 2 3. All right, so we have this.

[00:10:50]5,000 minus 4,000 gives you 1,000. So 1,000 n is equal to Let's subtract these numbers. What do we have there?

[00:11:01]Um we have um 996,000 minus 795,000 That's about uh Yeah, that's about 201 201 1 thousand. That's what you have there.

[00:11:19]996,000 minus 795,000. Okay.

[00:11:24]To get n, divide here by 1,000.

[00:11:28]Divide here by 1,000.

[00:11:33]So this would cancel this. We have that n is equal to 1 2 3 cancels 1 2 3.

[00:11:40]You're left with 201.

[00:11:43]So therefore Therefore, put it in statement, you say therefore um their salary will be equal after 201 years. Therefore the salary of A and B will be equal after After what's there? 201 years.

[00:12:19]So that's your answer, all right? That That's very crazy to even think that somebody has to work for 201 years to have equal salaries.

[00:12:27]It's crazy, but then that's your answer, okay?

[00:12:30]So this becomes the solution to this question, all right? So that's how you solve this question. All right. So if you enjoyed this video, don't forget to as usual hit the like button. Like this video. Oh, before I forget, let me give you a task.

[00:12:43]Let me give you a task for you to solve.

[00:12:46]So just hang on a bit. Let me try this question for yourself. The question here says, "Two men, A and B, started job at the same time.

[00:12:54]A at an annual salary of 4,000 naira with an annual increment of 1,000 naira and B at an annual salary of 5,000 naira with an annual increment of 500 naira.

[00:13:12]In which year of their services will their monthly gross pay be the same? In what year of their service will their monthly gross pay be the same? Also, after how many years or after how many years will they earn the same amount?

[00:13:30]All right? So that's the question. Now you try to answer this question. Pause this video. Answer this question. Leave your answer in the comment section. All right? All right, guys. So if you enjoyed this video as usual, don't forget to hit the like button. Give this video a thumbs up. Don't forget to also leave a comment. I gave you a task.

[00:13:50]Solve that task, all right? Pause this video right now. Solve that question.

[00:13:54]Leave your answer in the comment section and I'll give you a reply. I will tell you if you're correct or not. Don't forget to also subscribe. If it's your first time and you're yet to subscribe, please do well to join our community.

[00:14:05]Subscribe. Hit the bell icon and select all so that you get notified whenever we upload new content. Then finally, do well to share this video to your friends and your colleagues so that they can also learn. Thank you and see you in our next class.

[00:14:29]>> [cough and clears throat]