-3
  1. 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.

nicetomeetyou98
  • 13
  • 2
    What did you try? – Laxmi Narayan Bhandari Aug 12 '21 at 07:23
  • As the common difference of the arithmetic sequence is 1, it means that a1, a2, a3, a4 is differentiated by a, a+d, a+2d, a+3d respectively. For geometric sequence, it has a ratio of 3. It means that a1, a2, a3, a4 is differentiated by a, ar, ar^2, ar^3 respectively. Since the ratio (r) is 3, I substituted into the value which becomes a, 3a, 9a, 27a respectively. Since the 2nd and 4th terms adds up to 12 and 86, I came up with 2 equations which is 4a + d = 12 and 28a + 3d = 86. However, I found a and b but it is not the same as the answer so I am stuck. – nicetomeetyou98 Aug 12 '21 at 07:29
  • 1
    Common difference is 1, means d=1 – ACB Aug 12 '21 at 07:41
  • 1
    @YJ98 Applying, d =1 and the need for two different a's, a_1 for the arithmetic sequence and a_2 for the geometric sequence, what equations do you get?

    Also, be careful using the word differentiated. In Calculus, you'll spend a couple of months learning a whole different meaning for differentiation.

     – nickalh Aug 12 '21 at 08:58
  • Why has this question been downvoted? It's simpler than most here, but valid. Maybe his attempt to solve it should be moved from a comment to the actual question, but it appears quite valid in my eyes. – nickalh Aug 13 '21 at 06:28

1 Answers1

1

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.

nickalh
  • 1,263