Sure, let's start with the arithmetic sequence.
The terms of an arithmetic sequence can be represented by the formula: a_n = a_1 + (n-1)d where a_n is the nth term, a_1 is the first term, n is the position of the term in the sequence, and d is the common difference of the arithmetic sequence.
We know the first term (a_1) and the common difference (d) can be found by the formula:
d = (a_n - a_1) / (n - 1)
or:
a_1 = a_n - (n - 1)d
Let's denote the terms that we got as a_{35}=27 and a_{89}=4.
We can calculate the common difference using both of our possible formulas, let me show you the first way:
d = (a_{89}-a_{35}) / (89 - 35)
= (4 - 27) / (89 - 35)
= -23 / 54
= -0.426 (rounded to three decimal places)
Now going for the first term:
a_1 = a_{35} - (35 - 1) * d
= 27 - (34 * -0.426)
= 27 - (-14.48)
= 41.48
Hence, the common difference (d) is approximately -0.426 and the first term (a_1) is approximately 41.48.
Now let's address the geometric sequence.
The terms of a geometric sequence can represent as a_n = a_1 * r^(n-1) where a_n is the nth term, a_1 is the first term, n is the position of the term in the series, and r is the ratio of the sequence.
We know one term as a_{6}= -30 and the common ratio r is given as -1.
We can find the first term using the formula:
a_1 = a_n / r^(n-1)
= -30 / (-1)^(6-1)
= -30 / (-1)
= 30
Hence, the first term of the geometric sequence is 30.