new sequence formed by adding together corresponding term of a geometric sequence (G.S) and an arithmetic sequence (A.S).

a new sequence is formed when adding corresponding terms of a geometric sequence and an arithmetic sequence. The G.S has a common ratio of 3 and the A.S a common difference of -2. first two terms of the new sequence are 4 and 20. calculate term 3 of the new sequence...please help

If I well understand your question the geometric progression is: $$a\;,\;3a\;,\;9a\;,\;27 a\;,\;\cdots$$ and the arithmetic progression is $$b\;,\;b-2\;,\;b-4\;,\;b-6\;,\;\cdots$$

so, adding the corresponding terms $a_1+b_1$ and $a_2+b_2$ we have

$$a+b=4 \qquad 3a+b-2=20$$

solving the system of the two equation you can find $a,b$ and the third term.

If $a_1, a_2$ are the first two numbers in the arithmetic sequence and $g_1, g_2$ are the first two numbers in the geometric sequence. We have 4 unknowns. Can we create 4 equations based on the given information, and use that to solve for the 4 unknowns?

$a_1+g_1 = 4; a_2+g_2 = 20; a_2 = a_1 - 2; g_2 = 3 g_1$

That is enough information to solve for $a_1,g_1$ and you have the rules to find the next terms in the each sequence.