The first, third and ninth terms of a linear sequence (AP) are the first three terms of an exponential sequence (GP). If the seventh term of the linear sequence is 14,calculate:
(1) The twentieth term of the linear sequence;
(2) The sum the first twelve terms of the exponential sequence
Let the linear sequence be represented by the formula: a + nd, where a is the first term, d is the common difference, and n is the term number.
The given information tells us:
- First term of the linear sequence = a
- Third term of the linear sequence = a + 2d
- Ninth term of the linear sequence = a + 8d
- Seventh term of the linear sequence = a + 6d = 14
From this, we can find a = 14 - 6d.
Since the first three terms of the linear sequence are also the first three terms of a geometric sequence, we have:
a + nd = ar^(n-1)
a + 2d = ar
a + 8d = ar^8
Substitute the value of a from above, we get:
14 - 6d + nd = (14-6d)r^(n-1)
14 - 6d + 2d = (14-6d)r
14 - 6d + 8d = (14-6d)r^8
Now we have 3 equations with 3 variables. We can solve for d, r, and n.
Solving these equations would give us:
(1) The twentieth term of the linear sequence;
(2) The sum of the first twelve terms of the exponential sequence