The sum of the first two terms of a gp is 28, but the sum of the 3rd and 4th terms is 252. find the 2nd term of the GP if the 7th term is 5103.
a+ar = 28
ar^2+ar^3 = 252
solve for a and r, and then choose the values which make ar^6 positive
To find the second term of the geometric progression (GP), we will use the given information about the sum of the first two terms and the sum of the third and fourth terms.
Let's assume that the first term of the GP is 'a' and the common ratio is 'r'.
Given:
The sum of the first two terms (a + ar) is 28.
Therefore, we have the equation:
a + ar = 28. --- Equation 1
The sum of the third and fourth terms (ar^2 + ar^3) is 252.
Therefore, we have the equation:
ar^2 + ar^3 = 252. --- Equation 2
We are also given that the 7th term of the GP is 5103.
Therefore, we have the equation:
a * r^6 = 5103. --- Equation 3
Now, we have three equations with three variables (a, r).
To find the second term (ar), we first need to solve equations 1 and 2 simultaneously to find the values of 'a' and 'r'.
Let's solve equations 1 and 2:
From equation 1, we can rewrite it as a = 28 - ar.
Substituting a in equation 2, we get:
(28 - ar)r^2 + (28 - ar)r^3 = 252.
Simplifying this equation, we have:
28r^2 - ar^3 + 28r^3 - a r^4 = 252.
28r^2 - ar^3 + 28r^3 - ar^4 = 252.
28r^2 + 28r^3 - ar^3 - ar^4 = 252.
Now, combining like terms on the left side, we have:
28r^2 + 28r^3 - ar^3 - ar^4 - 252 = 0.
Let's simplify this equation further:
28r^2 + r^3(28 - a) - r^4(a) - 252 = 0.
28r^2 + (28 - a)r^3 - ar^4 - 252 = 0.
We can't directly solve this equation as there are two variables (a and r) in it. However, we have Equation 3, which we can use to substitute 'a' in terms of 'r'.
From Equation 3, we have: a = 5103 / r^6.
Substituting this value of 'a' in the equation 28r^2 + (28 - a)r^3 - ar^4 - 252 = 0, we get:
28r^2 + (28 - (5103 / r^6))r^3 - (5103 / r^6)r^4 - 252 = 0.
Simplifying this equation further, we get:
28r^2 + (28r^3 - 5103r) / r^6 - (5103r^4) / r^6 - 252 = 0.
Combining the fractions with common denominators, we get:
28r^2 + (28r^3 - 5103r - 5103r^4) / r^6 - 252 = 0.
Multiplying both sides by r^6 to eliminate the fractions, we have:
28r^8 + 28r^3 - 5103r - 5103r^4 - 252r^6 = 0.
This is a polynomial equation in terms of 'r'. To solve this equation to find the possible values of 'r', we can use numerical methods such as graphing or finding approximate solutions using calculator or mathematical software.
Once we find the value(s) of 'r', we can substitute it back to either Equation 1 or Equation 3 to find the corresponding value of 'a'.
However, please note that finding the second term might involve more complex calculations. If you have specific values for 'a', 'r', or any other terms in the GP, please provide them, and I can give you a more accurate and simplified answer.
To find the second term of the geometric progression (GP), we need to use the given information about the sum of terms.
Let's denote the first term of the GP as "a" and the common ratio as "r".
According to the problem statement, the sum of the first two terms is 28. So, we can write the equation as:
a + ar = 28 (Equation 1)
Similarly, the sum of the third and fourth terms is given as 252:
ar^2 + ar^3 = 252 (Equation 2)
Now, let's solve the first equation to find the value of a:
a + ar = 28
a(1 + r) = 28
a = 28 / (1 + r) (Equation 3)
Next, substitute this value of a in Equation 2:
(ar^2) + (ar^3) = 252
(r^2 + r^3) * a = 252
(r^2 + r^3) * (28 / (1 + r)) = 252 (Substituting the value of a from Equation 3)
Solving this equation will help us find the value of r, which is the common ratio.
Now, let's simplify the equation further:
(r^2 + r^3) * (28 / (1 + r)) = 252
28 * (r^2 + r^3) = 252 * (1 + r)
4 * (r^2 + r^3) = 36 * (1 + r)
r^2 + r^3 = 9 * (1 + r)
r^3 + r^2 - 9r - 9 = 0
To find the value of r, we need to solve this cubic equation.
Now, once we find the value of r, we can substitute it back into Equation 3 to find the value of "a".
Finally, knowing the values of "a" and "r", we can determine the second term of the GP by calculating:
Second term = a * r.