- The common difference of an arithmetic sequence is 1, and the common ratio of a geometric sequence is 3. A new sequence is formed by adding the corresponding terms of these two sequences. Suppose that the second and fourth terms of the new sequence are 12 and 86 respectively.
i) Find the nth term of the new sequence.
ii) Find the sum of the first n terms of the new sequence.
How about we use the following for the arithmetic terms
a, a + 1, a + 2, a + 3, ...
and the following for the geometic terms
b, 3b, 9b, 27b, ... ?
So the new sequence becomes
a+ b, a+1 + 3b, a +2 + 9b, a + 3 + 27b, ...
a + 1 + 3b = 12 and
a +2 + 9b = 86
So for part i, what's left is to solve the two related equations for a and b. Then write the formula for the nth term of the new sequence.
For part ii, A. separate the sequences back out,
B. Write the formula for sum of the arithmetic or addition sequence using n.
C. Write the formula for sum of the geometric or multiplication sequence using n.
D. Add C and D together.