What is a1 if a33 is equal to -2p and a74 = 26p?

Our topic is arithmetic series and sequence. Thanks.

To find the value of a1 in an arithmetic sequence, you need to know the common difference (d) and the values of any two terms in the sequence. In your case, you're given the values of a33 and a74, as well as their corresponding expressions in terms of p.

In an arithmetic sequence, the general formula to find the nth term (an) is:

an = a1 + (n - 1)d

where a1 is the first term, n is the term number, and d is the common difference.

Given that a33 = -2p and a74 = 26p, we can determine the value of the common difference (d) by subtracting the two equations:

a74 - a33 = (a1 + 73d) - (a1 + 32d)
26p - (-2p) = 99d

Simplifying further:

28p = 99d

Now, we know the value of d. The next step is to use a33 or a74 (both will give the same result) along with the common difference to find a1.

Using a33 = -2p:

-2p = a1 + (33 - 1)d
-2p = a1 + 32d

Substituting 28p/99 for d:

-2p = a1 + 32(28p/99)

Simplifying further:

-2p = a1 + (896p/99)

To isolate a1, we can add 2p to both sides:

a1 = -2p - (896p/99)

Combining the terms:

a1 = (-2p)(99/99) - (896p/99)
a1 = (-198p - 896p)/99
a1 = (-1094p)/99

Therefore, a1 = -1094p/99.

In summary, using the given values of a33 = -2p and a74 = 26p, we can determine that a1 in the arithmetic sequence is equal to -1094p/99.