If the first, third and fifth term of the linear sequence are the first three term of an exponential sequence. if the seventh term of the linear sequence is 14. calculate the twentieth term of the linear sequence
a+6d = 14
(a+2d)/a = (a+4d)/(a+2d)
solving that, we get a=14, d=0
that means the sequence is just constant
You sure there's no typo?
To solve this problem, we first need to find the common difference (d) of the linear sequence. We can do this by subtracting the third term from the first term.
Let's assume the first term of the linear sequence is 'a', and the common difference is 'd'. Therefore, the third term would be 'a + 2d', and the fifth term would be 'a + 4d'.
We are given that the first, third, and fifth terms of the linear sequence form the first three terms of an exponential sequence. This means that the ratio (r) between the third and first term is equal to the ratio between the fifth and third term.
The ratio (r) can be calculated by dividing the fifth term by the third term:
r = (a + 4d) / (a + 2d)
Since the first, third, and fifth terms form an exponential sequence, we know that:
r = (a + 4d) / (a + 2d) = (a + 2d) / a
Cross-multiplying, we get:
(a + 4d) * a = (a + 2d) * (a + 2d)
Expanding the equation:
a^2 + 4ad = a^2 + 4ad + 4d^2
Simplifying, we find that 4d^2 = 0, which implies that d = 0.
Now, we have the common difference (d) of the linear sequence, which is 0. This means that the sequence is constant, and each term will be the same.
We know that the seventh term is equal to 14. Since the common difference (d) is 0, the seventh term will be the same as the first term, which is 'a'. Therefore, a = 14.
Finally, to find the twentieth term of the linear sequence, we just need to use the value of 'a' (which is 14):
20th term = a = 14.
Thus, the twentieth term of the linear sequence is 14.