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### Course: Algebra 1 (Eureka Math/EngageNY)>Unit 3

Lesson 4: Topic A: Lessons 1-3: Geometric sequence formulas

# Sequences word problems

Sal solves two word problems about modeling real-world situations with arithmetic and geometric sequences.

## Want to join the conversation?

• How is it 1.4?? shouldn't it be 0.4??
• The number of leaves is 40% more than the number of leaves in the previous year, not 40% of the number of leaves in the previous year. Increasing a number by 40% means adding 100% of the number plus 40% of the number, which gives a total of 140% of the number. So the number of leaves is multiplying by 1.4 each year.
• How come Sal writes 50-3(n-1) instead of 50-(3)^n-1 like we have been learning for geometric sequences?
• Because the difference between consecutive term values is a fixed quantity, that means that the sequence is an arithmetic one!
• Technically couldn't you say you're adding 40%?
• Well, adding 40% of x is the same as x + 40%x, or x + 0.4x, or 1.4x. It's easier to just write it as 1.4x- clearer and more concise.
• If we were to write it as a function though, its form would be g(n) = 50-3n. So when she has 0 guests, she would have all the favors and when someone comes, he gets three favors right away. This premise of delayed bestowal of favors in this problem is a little confusing. If g(n) denotes the number of favors before the nth guest has arrived, then that means that the host is waiting for the next guest to come before giving the previous guest his favors and I don't see how the question supposes that. It explicitly says: "she gave 3 party favors to each of her guests as they arrived", so I guess "as" should mean "as soon as" which means "instantly".
• Giannis,

1. Expand the formula and simplify - it should be 53 - 3n. Now the maths makes sense.

2. To avoid repetition, read the commentary between Eric Allen Connor and David Severin. People are making this question way harder than it actually needs to be.

"as they arrived" would mean that 3 party favours were given when the person arrived at the door.

The 1st guest comes to the party and gets 3 party favours. Seo-Yun is left with 47 at the moment. When the 2nd guest comes, they get 3 as well. The 1st guest doesn't have to wait for the 2nd guest. Don't over complicate the question.
• 40% more = 1.4
40 % less = -1.4
40% multiplied = 0.4

That's the way to do it, right?
• Not quite 40% more is 1 + . 4 = 1.4 * __correct
40 % less is 1 - .4 = .6 * __

40% of (which is better wording than multiplied, but means the same) = .4 * ___
40% off of is same as 40% less
2 out of 3 ain't bad
• what is the difference between arithmetic and geo?
• An arithmetic sequence is where each new term is created by adding/subtracting a common value. A geometric sequence is where each term is created by multiplying/dividing by a common value.

Hope this helps.
• how did he get 1.4 out of 40% please?
• 40% more is the same thing as multiplying a number by 1.4 for example: 100 x 1.4 = 140 100 + 40% of 100 = 140. Did this help?
• why was 1.4 multiplied with 500 and even with the next no
• At Sal says “to grow by 40% you’re going to multiply by 1.4." Okay, so why by 1.4? I’ll break it down and use 500 for my example.

Beginning the first year there were 500 leaves. We can determine that the count of 500 leaves would increase by 200 more leaves (40% of 500 = 200; 0.4 * 500 = 200) during the first year.

Okay. At the end of the first year (i.e. the start of the second year) we will have 500 (our initial count) plus 200 (what grew during the year). So we get:

500 + 200 [The amount of leaves when starting the second year]
500 + ( 0.4 * 500 ) [We calculated 200 with 0.4 * 500 (40% of 500)]

I have two terms that both have 500 so I can factor the 500 out of both. To help illustrate that I will make an equivalent statement:

( 500 * 1 ) + ( 500 * 0.4 ) [ 500 * 1 is 500. ]

Now I can clearly use the distributive property to factor 500 out of both terms giving me:

500 * ( 1 + 0.4 )
500 * 1.4 [adding the two terms]

That’s how we get 1.4.

Now the second part of your question. Why do we multiply the next year’s number of leaves by 1.4? [Implying multiplying each of the following years.]

Simply put each year’s number of leaves grows an additional 40% from the previous year’s total of leaves. Just as we multiplied the first year by 1.4, we need to do it to the second year, the third year, … as the number of leaves grows each year.

I hope this helps.