Find the sum to 5 terms of the geometric progression whose first term is 54 and fourth term is 2
nth term of a geometric:
an = a1 rⁿ ⁻ ¹
a1 = initial value
r = common ratio
In this case:
a1 = 54
a4 = 2
a4 = a1 r⁴ ⁻ ¹ = 2
54 ∙ r³ = 2
r³ = 2 / 54
r³ = 2 / 2 ∙ 27
r³ = 1 / 27
r = ∛ (1 / 27 )
r = ∛1 / ∛27
r = 1 / 3
Sn = Sum first n terms of GP
Sn = a1 ( 1 - rⁿ ) / ( 1 - r )
S5 = Sum first 5 terms of GP
S5 = a1 ( 1 - r⁵ ) / ( 1 - r )
S5 = 54 [ 1 - ( 1 / 3 )⁵ ] / ( 1 - 1 / 3 )
S5 = 54 [ 1 - ( 1⁵ / 3⁵ ) ] / ( 3 / 3 - 1 / 3 )
S5 = 54 ( 1 - 1 / 243 ) / ( 2 / 3 )
S5 = 54 ( 243 / 243 - 1 / 243 ) / ( 2 / 3 )
S5 = 54 ( 242 / 243 ) / ( 2 / 3 )
S5 = ( 54 ∙ 242 / 243 ) / ( 2 / 3 )
S5 = 54 ∙ 242 ∙ 3 / 243 ∙ 2
S5 = 39204 / 486
S5 = 162 ∙ 242 / 162 ∙ 3
S5 = 242 / 3
Proof:
a1 = 54
a2 = 54 / 3 = 18
a3 = 18 / 3 = 6
a4 = 6 / 3 = 2
a5 = 2 / 3
a1 + a2 + a3 + a4 + a5 = 54 + 18 + 6 + 2 + 2 / 3 =
80 + 2 / 3 = 240 / 3 + 2 / 3 = 242 / 3
Well, well, well, let's get our calculators and funny bones ready! We have a geometric progression with the first term as 54 and the fourth term as 2.
To find the common ratio, let's divide the fourth term by the first term: 2/54, but hold on, we can simplify that to 1/27, which is our common ratio.
Now, we want to find the sum to 5 terms of this geometric progression. We know the first term is 54, so let's bust out our formula for the sum of a geometric series:
Sum = a(1 - r^n) / (1 - r)
In this case, a is 54 (our first term), r is 1/27 (the common ratio), and n is 5 (since we want the sum to 5 terms).
Now let's plug those values in and see what we get!
Sum = 54(1 - (1/27)^5) / (1 - (1/27))
Calculating, calculating... hmmm... yes... the suspense is killing me! And the sum of the first 5 terms of this geometric progression is approximately 54.036 (rounded to 3 decimal places).
So there you have it, after a series of clown calculations, the sum of the first 5 terms is approximately 54.036. Happy math-ing!
To find the sum of the first 5 terms of a geometric progression, we'll start by finding the common ratio (r) and the formula for the sum of an n-term geometric progression.
We have the first term, a₁, as 54.
We also have the fourth term, a₄, as 2.
The general formula for the nth term of a geometric progression is:
aₙ = a₁ * r^(n-1)
We can use this formula to find the common ratio (r).
Substituting the necessary values, we have:
2 = 54 * r^(4-1)
2 = 54 * r³
Dividing both sides of the equation by 54, we get:
2/54 = r³
1/27 = r³
Cube rooting both sides, we get:
r = ∛(1/27)
r = 1/3
Now that we have the common ratio, we can find the sum of the first 5 terms using the formula:
S₅ = a₁ * (1 - r⁵) / (1 - r)
Substituting the values we have, we get:
S₅ = 54 * (1 - (1/3)⁵) / (1 - 1/3)
Calculating the value of S₅, we get:
S₅ = 54 * (1 - 1/243) / (2/3)
S₅ = 54 * (242/243) * (3/2)
S₅ = 54 * 11/9
S₅ = 594/9
S₅ = 66
Therefore, the sum of the first 5 terms of the geometric progression is 66.
To find the sum of the first 5 terms of a geometric progression, you first need to find the common ratio. The common ratio (r) can be found by dividing any term in the progression by the previous term.
Given that the first term (a) is 54 and the fourth term (t4) is 2, we can find the common ratio as follows:
r = t4 / a
r = 2 / 54
To find the sum of the first 5 terms (S5), we can use the formula:
S5 = a * (1 - r^5) / (1 - r)
Substituting the values we know:
S5 = 54 * (1 - (2 / 54)^5) /(1 - (2 / 54))
Now we can calculate the common ratio and the sum of the first 5 terms of the geometric progression.