Find the sum of the G.p.2+6+18+54+...+1458
a = 2
r = 3
2*729 = 1458
729 = 3^6, so
S7 = 2(3^7-1)/(3-1) = 2*2186/2 = 2186
Solve the equation simultaneously.1/3x+y=3,x+1/2y=4
To find the sum of a geometric progression (G.P.), we need to use the formula for the sum of the first n terms:
Sn = a * (r^n - 1) / (r - 1)
In this case, we have the first term (a = 2) and the common ratio (r = 6/2 = 3).
The last term in the given G.P. is 1458. So, we need to find the value of n for which a * r^(n-1) = 1458.
Initially, we have:
2 * 3^(n - 1) = 1458
To solve for n, we can simplify the equation by dividing both sides by 2:
3^(n - 1) = 729
Now, we can take the logarithm of both sides of the equation to solve for n:
log(3^(n - 1)) = log(729)
(n - 1) * log(3) = log(729)
Using the logarithmic identity log(x^y) = y * log(x), we can rewrite the equation as:
(n - 1) = log(729) / log(3)
Evaluating the right side of the equation using a calculator, we get:
(n - 1) ≈ 6.018
Adding 1 to both sides, we find:
n ≈ 7.018
Since n must be a whole number, we can round n up to 8.
Now that we know the value of n, we can substitute it into the sum formula to find the sum of the G.P.:
Sn = a * (r^n - 1) / (r - 1)
S8 = 2 * (3^8 - 1) / (3 - 1)
Evaluating this expression, we can find the sum of the given G.P.:
S8 ≈ 1962