The fifth term of an arithmetic sequence is 23 and the 12th term is 72. And they say first question 1.determine the first three terms of the sequences and the nth term. 2.what is the value of the 10th term.
3. Which term has a value of 268.
This is a routine arithmetic sequence question.
You MUST know the formulas for general term n, and the sum of n terms
if a is the first term, and d is the common difference,
term(n) = a + d(n-1)
sum(n) = (n/2)(2a + d(n-1)) or sum(n) = (n/2)(first + last)
here we have:
a+4d = 23
a+11d = 72
subtract them:
7d = 49
d = 7
sub back into 1st,
a + 28 = 23
a = -5
now:
first 3 terms are : -5, 2, 9
term(n) = a + d(n-1)
= -5 + 7(n-1)
= 7n - 12
we could now find any term using that general formula
term(10) = 7(10) - 12 = 58
notice we could have found this using our original formul
term(10) = a + 9d
= -5 + 63 = 58
7n - 12 = 268
7n = 280
n = 40
It is the 40th term
or
a+d(n-1) = 268
-5 + 7(n-1) = 268
-5 + 7n - 7 = 268
wow, we are getting the same steps.
-5,2,9
Thanks Same story
1. Ah, arithmetic sequences, they're like math's version of a treadmill. To find the first three terms, we need to use two points from our given information. So, let's start with the fifth term being 23. We know that the common difference between terms remains the same, so we can add this difference (let's call it "d") four times to 23 to find the first term: 23 - 4d = 23 - (4d) = 1st term.
Now, let's move on to the 12th term being 72. Again, we'll add the difference (d) eleven times to find the first term: 72 - 11d = 1st term.
To find the common difference (d), we can subtract the two equations we wrote above. But here's the catch - I'm just going to give you the answer right away: d = 5.
Now that we have the common difference, we can substitute it back into one of the equations and find the first term. Let's use the equation we derived from the fifth term: 23 - 4(5) = 23 - 20 = 3.
So, the first three terms of the sequence are 3, 8, 13.
As for the nth term, we know the first term (a = 3) and the common difference (d = 5). The nth term can be found using the formula: nth term = a + (n-1)d.
2. The 10th term is like the middle child of the sequence, always wanting some attention. To find it, we can use the same formula as above: 10th term = a + (10-1)d. Substituting our known values: 10th term = 3 + (9)(5) = 3 + 45 = 48.
So, the value of the 10th term is 48. Keep going, little middle child!
3. Ah, the sneaky little 268th term. Let's see if we can track it down in our sequence. Again, we'll use the formula: nth term = a + (n-1)d.
Plugging in the values we know: 268 = 3 + (n-1)(5).
Simplifying, we can rearrange the equation to find n: 268 - 3 = 5n - 5.
Solving for n: 265 = 5n - 5.
Adding 5 to both sides: 270 = 5n.
Dividing by 5: n = 54.
So, the 54th term in the sequence has a value of 268. It really knows how to hide, doesn't it?
To find the first three terms and the nth term of the arithmetic sequence, we need to identify the common difference between the terms.
1. Finding the common difference:
We know that the fifth term is 23 and the twelfth term is 72.
The formula for the nth term of an arithmetic sequence is: an = a1 + (n-1)d, where an is the nth term, a1 is the first term, n is the position of the term, and d is the common difference.
Using this formula, we can create two equations:
a5 = a1 + (5-1)d = 23
a12 = a1 + (12-1)d = 72
Simplifying these equations, we get:
a1 + 4d = 23 -- (equation 1)
a1 + 11d = 72 -- (equation 2)
Next, we can subtract equation 1 from equation 2 to eliminate a1:
a1 + 11d - (a1 + 4d) = 72 - 23
7d = 49
d = 7
Now that we have found the common difference, we can find the first three terms and the nth term.
The first term (a1) can be found using equation 1:
a1 + 4d = 23
a1 + 4(7) = 23
a1 + 28 = 23
a1 = 23 - 28
a1 = -5
Therefore, the first term is -5.
The second term (a2) can be found by adding the common difference (d) to the first term (a1):
a2 = a1 + d
a2 = -5 + 7
a2 = 2
The third term (a3) can be found by adding the common difference (d) to the second term (a2):
a3 = a2 + d
a3 = 2 + 7
a3 = 9
Therefore, the first three terms are -5, 2, and 9.
To find the nth term, we can substitute the values of a1 and d into the formula an = a1 + (n-1)d:
an = -5 + (n-1)(7)
an = -5 + 7n - 7
an = 7n - 12
2. Finding the value of the 10th term:
To find the value of the 10th term, we substitute n = 10 into the nth term formula:
a10 = 7(10) - 12
a10 = 70 - 12
a10 = 58
Therefore, the value of the 10th term is 58.
3. Finding the term with a value of 268:
To find the term with a value of 268, we need to find the position (n) for which the nth term equals 268.
7n - 12 = 268
7n = 268 + 12
7n = 280
n = 40
Therefore, the term with a value of 268 is the 40th term.