The 15th and 21st terms of an arithmetic sequence are -67 and -97. What is the 30th term?

This is all I can figure out?

-97- (-24)= -30 common difference.

HELP!

but you have 6 terms in that range,

so the common difference
= d
= -5

or

use the formulas which you must have and know to do this topic
term15 = a+14d = -67
term21 = a + 20d = -97
subtract them :
6d = -30 , see above
d = -5

in a+14d = -67
a - 70 = -67
a = 3

term 30 = a + 29d
= 3 + 29(-5) = -142

Well, it seems like your math skills are in a bit of a circus act right now. Don't worry, I'll clown around with the numbers and help you out!

To find the common difference in an arithmetic sequence, we subtract the 15th term (-67) from the 21st term (-97):

-97 - (-67) = -97 + 67 = -30

So, you've got the correct common difference of -30! Great job on that part.

Now, to find the 30th term, we need to determine the pattern. Since the common difference is -30, we can keep subtracting it from the 21st term until we reach the 30th term:

-97 - 30 = -127
-127 - 30 = -157
And so on...

But hey, instead of doing all that math, let's summon our funny bone and figure it out. How do you make seven even? Just take away the "s"! So, if we take away the 30th term (the "s") from 21 (the current term), we get:

21 - 30 = -9

Therefore, the 30th term of the arithmetic sequence is -9. I hope this circus act of mathematics brought a smile to your face!

No worries! Let's use the formula for the nth term of an arithmetic sequence to find the 30th term.

The formula for the nth term of an arithmetic sequence is:
a_n = a_1 + (n - 1)d

Given information:
a_15 = -67
a_21 = -97
d = -30 (common difference)

We can use these values to find a_1, the first term of the sequence.

To find a_1, we'll use the formula for the nth term but with n = 15:
a_15 = a_1 + (15 - 1)d

We substitute the given value of a_15 and d into the equation:
-67 = a_1 + (15 - 1)(-30)

Now, simplify and solve for a_1:
-67 = a_1 + 14(-30)
-67 = a_1 - 420
a_1 = -67 + 420
a_1 = 353

Now that we have the first term, a_1, we can find the 30th term, a_30, by plugging the values into the nth term formula:
a_30 = 353 + (30 - 1)(-30)

Simplify and calculate:
a_30 = 353 + 29(-30)
a_30 = 353 - 870
a_30 = -517

Therefore, the 30th term of the arithmetic sequence is -517.

To find the 30th term of an arithmetic sequence, we need the common difference (d) and the first term (a₁).

In this case, we are given the 15th and 21st terms:
a₁₅ = -67 and a₂₁ = -97.

We can use these terms to find the common difference:

d = a₂₁ - a₁₅
d = -97 - (-67)
d = -97 + 67
d = -30

Now that we have the common difference, we can find the first term (a₁) using any of the given terms. Let's use a₁₅:

a₁₅ = a₁ + (15-1)d
-67 = a₁ + 14(-30)
-67 = a₁ - 420
a₁ = -67 + 420
a₁ = 353

Now we have the first term (a₁ = 353) and the common difference (d = -30), we can find the 30th term (a₃₀):

a₃₀ = a₁ + (30-1)d
a₃₀ = 353 + 29(-30)
a₃₀ = 353 - 870
a₃₀ = -517

So, the 30th term of the arithmetic sequence is -517.