1. Examine the first four terms in each of the following number sequences:
*2, 7, 12, 17 my formula x+5
*7, 21, 63, 189, 567 my formula x(2)+x
*1000, 850, 700, 550 my formula x-150
* –10, –6, –2, 2 my formula x+4
For each sequence, what would be the 20th and 100th term? Explain your reasoning for each.
For which of the sequences would the number 352 be a term? Explain your reasoning.
One of the sequences is different from the other three. Describe what makes the one different and the other three "alike."
All your choices are correct except the 2nd
look at the terms:
7, 21, 63, 189, 567, ...
or
7*1, 7*3, 7*9, 7*27, 7*81
looks like 7* 3^(x-1)
btw, that sequence is the one that different from all the others, because it is NOT an arithmetic sequence.
For the 20th and 100th term, I will do the first, you repeat the same method for the others
for 2 7 12 17 ...
a = 2, d = 5
term 20 = a + 19d
= 2 + 19(5) = 97
term 100 = a + 99d
= 2 + 99(5) = 497
As to 352 being one of the terms?
let the term number be n
term n = a + (n-1)d
2 + (n-1)(5) = 352
5n - 5 = 350
5n = 355
n = 71
yes, 352 is the 71st term in the first sequence
repeat for the 3rd and 4th sequence
testing for the 2nd sequence
7*(3)^(n-1) = 352
3^(n-1) = 352/7
which clearly is NOT a power of 3, so 352 does not belong to the 2nd sequence
7*1, 7*3, 7*9, 7*27, 7*81
looks like 7* 3^(x-1) I don't understand how you got (x-1) can you explain?
The multipliers are
1,3,9,27, ...
As powers of 3, those are 0,1,2,3,...
You can't say 7*3^x because the first term (x=1) has not been multiplied yet.
To find the 20th and 100th term for each sequence, we can use the given formulas.
1. Sequence: 2, 7, 12, 17
Formula: x + 5
To find the 20th term, substitute x = 2 in the formula:
20th term = 2 + 5 × (20 - 1) = 2 + 5 × 19 = 2 + 95 = 97
To find the 100th term, substitute x = 2 in the formula:
100th term = 2 + 5 × (100 - 1) = 2 + 5 × 99 = 2 + 495 = 497
2. Sequence: 7, 21, 63, 189, 567
Formula: x(2) + x
To find the 20th term, substitute x = 7 in the formula:
20th term = 7(2) + 7 × (20 - 1) = 7 × 2 + 7 × 19 = 14 + 133 = 147
To find the 100th term, substitute x = 7 in the formula:
100th term = 7(2) + 7 × (100 - 1) = 7 × 2 + 7 × 99 = 14 + 693 = 707
3. Sequence: 1000, 850, 700, 550
Formula: x - 150
To find the 20th term, substitute x = 1000 in the formula:
20th term = 1000 - 150 × (20 - 1) = 1000 - 150 × 19 = 1000 - 2850 = -1850
To find the 100th term, substitute x = 1000 in the formula:
100th term = 1000 - 150 × (100 - 1) = 1000 - 150 × 99 = 1000 - 14850 = -13850
4. Sequence: -10, -6, -2, 2
Formula: x + 4
To find the 20th term, substitute x = -10 in the formula:
20th term = -10 + 4 × (20 - 1) = -10 + 4 × 19 = -10 + 76 = 66
To find the 100th term, substitute x = -10 in the formula:
100th term = -10 + 4 × (100 - 1) = -10 + 4 × 99 = -10 + 396 = 386
For the sequence where 352 would be a term, we can plug it into each formula and see if it fits:
1. x + 5: 352 + 5 = 357 (not a match)
2. x(2) + x: 352(2) + 352 = 704 + 352 = 1056 (not a match)
3. x - 150: 352 - 150 = 202 (not a match)
4. x + 4: 352 + 4 = 356 (not a match)
None of the sequences have 352 as a term.
The one sequence different from the other three is: 7, 21, 63, 189, 567. This sequence follows the pattern of multiplying the previous term by 3, whereas the other three sequences have a constant addition or subtraction. The other three sequences are similar as they all have a fixed pattern involving either adding or subtracting a constant value.