If 3, p, q, 24 are consecutive terms of an exponential sequence find the values of p and q hence find the sum of the first seven terms
just remember there's a common ratio r. That means that
3r^3 = 24
Now find r, and then
S7 = 3(r^7-1)/(r-1)
To find the values of p and q, we can use the idea that consecutive terms of an exponential sequence have a common ratio.
Let's say the common ratio is r. In this case, we have:
3 * r = p (since p is the next term after 3)
p * r = q (since q is the next term after p)
q * r = 24 (since 24 is the next term after q)
To continue, we can use substitution. From the first equation, we have r = p/3. Substituting this into the second equation gives:
(p/3) * r = q
(p/3) * (p/3) = q
Now, we can substitute this value of q into the third equation:
q * r = 24
[(p/3) * (p/3)] * (p/3) = 24
Simplifying this equation will give us a value for p:
(p^3) / 27 = 24
p^3 = 24 * 27
p^3 = 648
Now, we can find the cube root of 648 to find the value of p:
p = ∛648
p = 8
Now that we have p, we can substitute it back into the previous equations to find the value of q:
q = (p/3) * (p/3)
q = (8/3) * (8/3)
q = 64/9
So, we have found the values of p and q. p = 8 and q = 64/9.
To find the sum of the first seven terms, we need to add up the values of the terms from 3 to q.
The sum of a geometric sequence can be found using the formula:
S = a * (1 - r^n) / (1 - r)
where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
In this case, the first term is 3, the common ratio is p/3 (or 8/3), and we want to find the sum of the first seven terms (n = 7).
Using the formula:
S = 3 * (1 - (8/3)^7) / (1 - (8/3))
Calculating this expression will give us the sum of the first seven terms.