Ryan is building matchstick square sequences, as shown ( first one used 4, second one used 7, third one used 10, and the forth one used 13). He used 599 matchsticks to form the last two figures in his sequences. How many matchsticks did he use in each of the last two figures?
298 and 301
To solve this problem, we need to find the pattern in the number of matchsticks used to form each figure in the sequence.
Looking at the given sequence, we can observe that the difference between each term is 3. So, we can conclude that the sequence follows an arithmetic pattern with a common difference of 3.
To find the number of matchsticks used in the fourth and fifth figures, we can use the arithmetic formula for the nth term of an arithmetic sequence:
Tn = a + (n - 1)d
Where Tn is the nth term, a is the first term, n is the term number, and d is the common difference.
In this case, we know that the fourth term uses 13 matchsticks, so a = 13. We want to find the fifth term, so n = 5. And since the common difference is 3, d = 3.
Using the formula, we can calculate the number of matchsticks used in the fifth term:
T5 = 13 + (5 - 1) * 3
= 13 + 4 * 3
= 13 + 12
= 25
Therefore, Ryan used 25 matchsticks in the fifth figure.
Now, to find the number of matchsticks used in the fourth figure, we need to subtract the number of matchsticks used in the fifth figure from the total number of matchsticks used to form the last two figures:
Total matchsticks used = Number of matchsticks used in fourth figure + Number of matchsticks used in fifth figure
Given that the total matchsticks used is 599 and the number of matchsticks used in the fifth figure is 25, we can substitute these values into the equation:
599 = Number of matchsticks used in fourth figure + 25
Solving for the number of matchsticks used in the fourth figure:
Number of matchsticks used in fourth figure = 599 - 25
= 574
Therefore, Ryan used 574 matchsticks in the fourth figure.