Question

For a certain arithmetic sequence, tle="a_{3} =14, a_{10}=98" alt="a_{3} =14, a_{10}=98" align="absmiddle" class="latex-formula">. Write the first three terms of the sequence.
I'll mark you as brainliest *only if you answer*. Thanks <3

Answer 1

Answer:

An arithmetic sequence is defined by:

$a_n = a_{n-1} + d$

Here, we know that:

$a_3 = 14\\a_{10} = 98$

Using the recursive relation, we can write:

$a_{10} = 98 = a_9 + d$

And can keep applying the recursive relation, so we will have:

$98 = a_9 + d = (a_8 + d) + d = a_8 + 2*d$

And so on, until we get to a known term:

$98 = a_8 + 2*d = a_7 + 3*d = a_6 + 4*d = a_5 + 5*d = a_4 + 6*d = a_3 + 7*d$

$98 = a_3 + 7*d = 14 + 7*d$

Then we can solve:

98 = 14 + 7*d

for d:

98 - 14 = 7*d

84 = 7*d

84/7 = d = 12

Now we know the value of d, we can keep using the recursive relation to find the first 3 terms:

Here, we can rewrite the recursive relation as:

$a_{n-1} = a_n - d$

Then:

$a_2 = a_3 - d = 14 - 12 = 2\\a_2 = 2$

$a_1 = a_2 - d = 2 - 12 = -10\\a_1 = -10$

$a_0 = a_1 - d = -10 - 12 = -22\\a_0 = -22$

Then the first 3 terms are:

-22, -10, 2