The Ratio Of The 2nd Term To The 7th Term Of An A.P Is 1:3 And Their Sum Is 20.Calculate The Sum Of The 1st Terms Of The G.P
2nd term = a+d
7th term = a+6d
a+d + a+6d = 20
2a + 7d = 20 **
(a+d)/(a+6d) = 1/3
3a + 3d = a + 6d
2a = 3d ***
sub that into **
3d + 7d = 20
10d = 20
d = 2
then 2a = 6
a = 3
"Calculate The Sum Of The 1st Terms Of The G.P"
HUH? I thought we were dealing with an AP.
Isn't the sum of the first term just the first term???
Let's begin by finding the common difference of the arithmetic progression (A.P).
The ratio of the 2nd term to the 7th term is given as 1:3. This means that the 2nd term is one-third of the 7th term.
Let the 2nd term be "a" and the 7th term be "d".
So, a : d = 1 : 3.
We can write this as a/d = 1/3.
Cross-multiplying, we get a = (d/3).
We are also given that the sum of these terms is 20. So,
a + d = 20.
Substituting the value of "a" from the previous equation, we have:
(d/3) + d = 20.
Multiplying through by 3, we get:
d + 3d = 60.
Combining like terms, we have:
4d = 60.
Dividing both sides by 4, we find:
d = 15.
Substituting this value of "d" back into the equation a + d = 20, we have:
a + 15 = 20.
Subtracting 15 from both sides, we get:
a = 5.
So, the 1st term of the arithmetic progression (A.P) is 5.
Now, let's calculate the sum of the 1st terms of the geometric progression (G.P).
The sum of the 1st terms of a geometric progression (G.P) is given by the formula:
S = a / (1 - r),
where "S" is the sum of the terms, "a" is the 1st term, and "r" is the common ratio.
But since we only know the sum of the terms in the arithmetic progression (A.P), we need to find the common ratio of the geometric progression (G.P).
To find the common ratio, we can divide the 7th term of the A.P (d) by the 2nd term of the A.P (a).
r = d/a = 15/5 = 3.
Substituting the values into the formula, we have:
S = 5 / (1 - 3).
Simplifying, we get:
S = 5 / (-2),
S = -2.5.
Therefore, the sum of the 1st terms of the geometric progression (G.P) is -2.5.
To solve this problem, we need to first understand what an A.P. (Arithmetic Progression) and G.P. (Geometric Progression) are.
An Arithmetic Progression is a sequence of numbers in which the difference between any two consecutive terms is constant. The first term is denoted as 'a' and the common difference is denoted as 'd'.
A Geometric Progression is a sequence of numbers in which the ratio between any two consecutive terms is constant. The first term is denoted as 'a' and the common ratio is denoted as 'r'.
Now, let's address the question at hand. We know that the ratio of the 2nd term to the 7th term of the Arithmetic Progression is 1:3. Let's assume the 2nd term is 'a' and the common difference is 'd'.
So, the 7th term will be:
7th term = 2nd term + (7-2) * common difference
=> 7th term = a + 5d
Given that the sum of these two terms is 20, we can write:
2nd term + 7th term = 20
a + (a + 5d) = 20
2a + 5d = 20 ---(Equation 1)
Now, let's find the sum of the 1st terms of the Geometric Progression. Let's assume the first term is 'a' and the common ratio is 'r'.
The sum of the first 'n' terms of a geometric progression is given by the formula:
Sum of 'n' terms = (a * (r^n - 1)) / (r - 1)
We are required to find the sum of the 1st terms of the Geometric Progression. Since this is not specified, we need to assume a value for 'n'. Let's assume 'n' is a positive integer.
Therefore, the sum of the first terms of the Geometric Progression is:
Sum of the 1st terms = (a * (r^n - 1)) / (r - 1) ---(Equation 2)
We cannot directly calculate the sum of the 1st terms of the Geometric Progression without knowing the values of 'a' and 'r'. Given no additional information, the problem is not sufficiently defined to calculate the sum of the first terms of the Geometric Progression.