A g.p has 6 terms. If the third and fourth term are 28 and_56 respectively find the first term, the sum of the g.p and the product of the tenths and 20 term
Just translate the English into Math.
"If the third and fourth term are 28 and_56 respectively "
ar^2 = 28
ar^3 = 56
divide the second equation by the first.
r = 56/28 = 2
from ar^2 = 28
a(4) = 28
a = 7
Sum(6) = a(r^6 - 1)/(r-1)
= 7(2^6 - 1)/1= 441
You said, the GP has 6 terms, why are are finding stuff about
term10 and term20??
anyway .....
term(10) = 7(2^9)
term(20) = 7(2^19)
product = 49(2^28) , write it in scientific notation or whatever way you want
To find the first term of the geometric progression (G.P.), we can use the formula for the nth term of a G.P:
Tn = a * r^(n-1)
where Tn is the nth term, a is the first term, r is the common ratio, and n is the number of terms.
In this case, we know the third term (T3) is 28, and the fourth term (T4) is 56.
Using the formula for T3 and T4:
T3 = a * r^(3-1) = 28
T4 = a * r^(4-1) = 56
Dividing the equation for T4 by the equation for T3:
T4 / T3 = (a * r^(4-1)) / (a * r^(3-1))
56 / 28 = r
This simplifies to:
2 = r
Now, we can substitute the value of r back into the equation for T3 to solve for the first term (a):
28 = a * 2^(3-1)
28 = a * 4
a = 28 / 4
a = 7
So, the first term of the G.P. is 7.
Next, to find the sum of the geometric progression, we can use the formula:
Sn = a * (r^n - 1) / (r - 1)
where Sn is the sum of the G.P., a is the first term, r is the common ratio, and n is the number of terms.
In this case, we know that there are 6 terms in the G.P., so:
n = 6
Substituting the values we know into the formula:
S6 = 7 * (2^6 - 1) / (2 - 1)
S6 = 7 * (64 - 1) / 1
S6 = 7 * 63
S6 = 441
So, the sum of the G.P. is 441.
Finally, to find the product of the tenths and 20th term, we can calculate the 20th term of the G.P. and multiply it by 1/10 (or 0.1):
T20 = a * r^(20-1)
T20 = 7 * 2^19
T20 = 7 * 524288
T20 = 3670016
Product = 3670016 * 0.1
Product = 367001.6
So, the product of the tenths and 20th term is 367001.6.