the second term of geometric sequence is 6 and the fifth term is _48. find also, in simplified form, of the first n terms of this sequence.
what is the answer
r^(5 - 2) = 48 / 6 ... r^3 = 8 ... r = 2
An = 3 * 2^(n-1)
Just use your definitions:
"the second term of geometric sequence is 6" ----> ar = 6
"the fifth term is - 48" ---> ar^4 = -48
divide them:
(ar^4)/(ar) = -48/6
solve for r, since the a's will divide out
then plug that back into ar=6
sub into the formula for the sum of n terms.
let me know what you get
Why did the geometric sequence go to therapy? Because it couldn't handle its exponential growth!
Now, let's solve your math problem.
First, let's find the common ratio (r) of the geometric sequence. To do that, we divide the fifth term (a₅ = -48) by the second term (a₂ = 6):
r = a₅ / a₂ = -48 / 6 = -8
Now we can find the first term (a₁) by using the formula:
a₁ = a₂ / r = 6 / -8 = -3/4
The formula for the n-th term of a geometric sequence is given by:
aₙ = a₁ * r^(n-1)
Now, let's simplify the formula for the sum of the first n terms of a geometric sequence:
Sₙ = a₁ * (1 - rₙ) / (1 - r)
Plug in the values we found:
Sₙ = (-3/4) * (1 - (-8)^n) / (1 - (-8))
Sₙ = (-3/4) * (1 + 8^n) / 9
So, in simplified form, the sum of the first n terms of this sequence is (-3/4) * (1 + 8^n) / 9.
And remember, math problems may seem tough, but they're nothing compared to a clown juggling with algebraic equations!
To find the common ratio of the geometric sequence, we can divide the fifth term by the second term.
Let's call the common ratio "r".
The second term is 6, so a2 = 6, and the fifth term is -48, so a5 = -48.
a5 = a2 * r^(5-2)
-48 = 6 * r^3
We can divide both sides of the equation by 6 to simplify it:
-8 = r^3
Now, we need to find the value of r. Taking the cube root of both sides, we have:
r = ∛(-8)
The cube root of -8 is -2, so r = -2.
Now that we know the common ratio, we can find the first term (a1) by dividing the second term (6) by the common ratio (-2):
a1 = a2 / r
a1 = 6 / -2
a1 = -3
So, the first term of the sequence is -3, and the common ratio is -2.
Now, let's find the formula for the nth term of the geometric sequence. For a geometric sequence, the formula is given by:
an = a1 * r^(n-1)
We have a1 = -3 and r = -2, so substituting these values into the formula, we get:
an = -3 * (-2)^(n-1)
To find the sum of the first n terms (Sn) of the sequence, we can use the formula:
Sn = a1 * (1 - r^n) / (1 - r)
Substituting in the values we know, we have:
Sn = -3 * (1 - (-2)^n) / (1 - (-2))
Simplifying further:
Sn = -3 * (1 - (-2)^n) / 3
The formula, in simplified form, for the sum of the first n terms of this geometric sequence is:
Sn = 1 - (-2)^n
To find the value of S_n, substitute the value of n into this formula.