Find the sum to 5 terms of the geometric progression whose first term is 54 and fourth term is 2.
what about the solution
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To find the sum of the first 5 terms of a geometric progression, we need to use the formula:
Sn = a * (1 - r^n) / (1 - r),
where:
- Sn is the sum of the first n terms,
- a is the first term,
- r is the common ratio,
- n is the number of terms.
In this case, we are given:
- a = 54 (first term),
- the fourth term = 2.
Let's use this information to find the common ratio (r) first:
Given that the fourth term (T4) is 2, we can express it in terms of the first term (a) and the common ratio (r):
T4 = a * r^(4-1)
2 = 54 * r^3
To solve for r, we can divide both sides of the equation by 54:
2/54 = r^3
1/27 = r^3
Taking the cube root of both sides, we find:
r = ∛(1/27)
r = 1/3
Now that we have the common ratio (r = 1/3), we can plug it into the formula to find the sum of the first 5 terms (Sn):
Sn = a * (1 - r^n) / (1 - r)
Substituting the given values:
Sn = 54 * (1 - (1/3)^5) / (1 - 1/3)
Evaluating the expression, we get:
Sn = 54 * (1 - 1/243) / (2/3)
= 54 * (242/243) / (2/3)
= 54 * 242 * 3 / (243 * 2)
= (54 * 242 * 3) / (486)
= 3924 / 486
= 8.07 (approximately)
Therefore, the sum of the first 5 terms of the given geometric progression is approximately 8.07.
a=54
r=1/3
Take it from there.