determine the 100th term of the arithmetic sequence with t7=465 and t13=219

t7=465 ---> a+6d=465

t13=219 --> a+12d=219
subtract:
6d = --246
d = -41
in 1st equation: a+6(-41) = 465
a = 711

term(100) = a+99d = ......

you finish it

term(100)= 711+99(-41)

term(100)= -3348

Well, if I had a dollar for every arithmetic sequence problem, I'd have... well, not a lot, but let's solve this anyway!

We know that in an arithmetic sequence, each term is obtained by adding a common difference to the previous term. Now, let's find the common difference (d).

We can use the formula for the nth term of an arithmetic sequence:

t(n) = a + (n - 1)d

Given that t(7) = 465, we can substitute n = 7:

465 = a + (7 - 1)d

Similarly, using t(13) = 219, we substitute n = 13:

219 = a + (13 - 1)d

Now, we have a nice little system of equations to solve. But fear not, because I'll do the heavy lifting!

From the first equation, we get:

d = (465 - a) / 6

Substituting this value for d in the second equation:

219 = a + (13 - 1)((465 - a) / 6)

Now, let's simplify and solve for a:

219 = a + 12((465 - a) / 6)

219 = a + 2(465 - a)

219 = a + 930 - 2a

219 - 930 = -a

-a = -711

a = 711

So, we found that the first term (a) is 711.

Now that we have a and d, we can use the formula for the nth term to find t(100):

t(100) = 711 + (100 - 1)d

Plugging in the values:

t(100) = 711 + 99d

But we don't know what d is... Ah, right! We found d earlier:

d = (465 - a) / 6

d = (465 - 711) / 6

d = -246 / 6

d = -41

Now, let's substitute that back into the nth term formula:

t(100) = 711 + 99(-41)

t(100) = 711 - 4059

t(100) = -3348

So, the 100th term of the arithmetic sequence is -3348. Now that's a negative milestone!

To determine the 100th term of an arithmetic sequence, we need to find the common difference (d) first.

Given the information that t7 = 465 and t13 = 219, we can use the formula for the nth term of an arithmetic sequence:

tn = a + (n-1)d

In this case, a refers to the first term, n is the term number, and d is the common difference.

Let's substitute the values of t7 and t13 into the formula:

t7 = a + (7-1)d = 465
t13 = a + (13-1)d = 219

We have a system of two equations:
a + 6d = 465 (Equation 1)
a + 12d = 219 (Equation 2)

Now we can solve this system of equations to find the values of a and d.

Subtracting Equation 2 from Equation 1 eliminates a:
(a + 6d) - (a + 12d) = 465 - 219
-6d = 246
d = -246/(-6)
d = 41

Now that we have d, we can use Equation 1 to find a:
a + 6d = 465
a + 6(41) = 465
a + 246 = 465
a = 465 - 246
a = 219

We have found that the first term (a) is 219 and the common difference (d) is 41.

Now we can find the 100th term (t100) using the formula:

t100 = a + (100-1)d
t100 = 219 + (99)41
t100 = 219 + 4059
t100 = 4278

Therefore, the 100th term of the arithmetic sequence is 4278.

To determine the 100th term of an arithmetic sequence, we need to find the common difference (d) first.

The formula to find the "nth" term (tn) of an arithmetic sequence is given by:

tn = a + (n - 1)d

Where:
tn is the "nth" term
a is the first term of the sequence
n is the position of the term
d is the common difference between consecutive terms

We can use this formula to find the common difference. Given that t7 = 465 and t13 = 219, we have:

t7 = a + (7 - 1)d
465 = a + 6d

t13 = a + (13 - 1)d
219 = a + 12d

Now we have a system of two equations with two variables (a and d):

Equation 1: 465 = a + 6d
Equation 2: 219 = a + 12d

Solving these equations simultaneously will give us the values of a and d:

Let's solve these equations:

By subtracting Equation 2 from Equation 1, we can eliminate 'a' and solve for 'd':

465 - 219 = (a + 6d) - (a + 12d)
246 = -6d
d = -246 / -6
d = 41

Now that we have the common difference (d = 41), we can find the first term (a) using either of the equations:

a = t7 - (7 - 1)d
= 465 - (6 × 41)
= 465 - 246
= 219

So, the first term (a) is 219.

Now, we can use the formula to find the 100th term (t100):

t100 = a + (100 - 1)d
= 219 + (99 × 41)
= 219 + 4059
= 4278

Therefore, the 100th term of the arithmetic sequence is 4278.