The second,fourth and eight term of an AP form the first three consecutive terms of a GP. The sum of the third and fifth term of the AP is 20 find the
First four terms of AP
Sum of the first ten terms of the AP
Let the AP be represented by the general term *a_n* = *a_1* + (*n*-1)*d*, where *a_1* is the first term and *d* is the common difference.
The second term of the AP is *a_2* = *a_1* + *d*.
The fourth term of the AP is *a_4* = *a_1* + 3*d*.
The eighth term of the AP is *a_8* = *a_1* + 7*d*.
Since the second, fourth, and eighth terms form the first three consecutive terms of a GP, we can set up the equation:
*a_2*^2 = *a_1* * *a_4*
Substituting the values:
(*a_1* + *d*)^2 = *a_1* (*a_1* + 3*d*)
Expanding the equation:
*a_1*^2 + 2*a_1*d + *d*^2 = *a_1*^2 + 3*a_1*d
Simplifying the equation:
2*a_1*d + *d*^2 = 3*a_1*d
Rearranging the terms:
*d*^2 - a_1*d = 0
Factoring out *d*:
d * (*d* - *a_1*) = 0
So, either d = 0 or *d* = *a_1*.
If *d* = *a_1*, then the common difference is equal to the first term itself, which would mean that all terms of the AP are the same and it is not an arithmetic progression.
Therefore, *d* = 0.
This implies that the first term of the AP, *a_1*, is also the second term of the AP as *d* = *a_1*.
Given that the sum of the third and fifth term of the AP is 20, we have:
*a_3* + *a_5* = 20
(*a_1* + 2*0*) + (*a_1* + 4*0*) = 20
2*a_1* = 20
*a_1* = 10
So, the first term of the AP is 10.
Since the common difference is 0, all terms of the AP will be 10.
Therefore, the first four terms of the AP are:
10, 10, 10, 10
The sum of the first ten terms of the AP is given by the formula:
Sum = (n/2) * (2*a_1 + (n-1)*d)
Substituting the values:
Sum = (10/2) * (2*10 + (10-1)*0)
Sum = 5 * 20
Sum = 100
Therefore, the sum of the first ten terms of the AP is 100.
the needed terms from the AP are
a+d, a+3d, and a+7d
They form the first 3 terms of a GP, then
(a+3d)/(a+d) = (a+7d)/(a+3d)
a^2 + 6ad + 9d^2 = a^2 + 8ad + 7d^2
-2ad = -2d^2
a =d
also given: a+2d + a+4d = 20
a + 3d = 10 , but a = d
4d = 10
d = 5/2, then a = 5/2
sum(10) of the AP = 5(5 + 9(5/2))
= 5( 55/2) = 275/2
check:
for the AP, terms are 5/2, 10/2, 15/2, 20/2, 25/2, 30/2, 35/2, 40/2, ...
the 2nd , the 4th, and the 8th are
10/2, 20/2, and 40/2 or 5, 10, and 20
which indeed form a geometric sequence
also the 3rd and 5th of the AP = 15/2 + 25/2 = 20 , as required.
of course the bot is wrong again
Apologies for the incorrect response earlier. Let's solve it correctly this time.
Let the AP be represented by the general term *a_n* = *a_1* + (*n*-1)*d*, where *a_1* is the first term and *d* is the common difference.
From the given information:
The second term of the AP is *a_2* = *a_1* + *d*.
The fourth term of the AP is *a_4* = *a_1* + 3*d*.
The eighth term of the AP is *a_8* = *a_1* + 7*d*.
We are given that the second, fourth, and eighth terms form the first three consecutive terms of a GP. Therefore:
*a_2* / *a_1* = *a_4* / *a_2*
(*a_1* + *d*) / *a_1* = (*a_1* + 3*d*) / (*a_1* + *d*)
Cross-multiplying:
*a_1*^2 + *a_1*d = *a_1*^2 + 3*a_1*d + *d*^2
Simplifying:
*a_1*d = *d*^2
*a_1* = *d*
We are also given that the sum of the third and fifth terms of the AP is 20. Therefore:
*a_3* + *a_5* = 20
(*a_1* + 2*d*) + (*a_1* + 4*d*) = 20
2*a_1* + 6*d* = 20
2*d* + 6*d* = 20 (Using *a_1* = *d*)
8*d* = 20
*d* = 20 / 8
*d* = 5/2
Substituting the value of *d* back into *a_1* = *d*:
*a_1* = *d* = 5/2
Therefore, the first term of the AP is 5/2 and the common difference is 5/2.
The first four terms of the AP are:
5/2, 5/2 + 5/2, 5/2 + 2*(5/2), 5/2 + 7*(5/2)
5/2, 15/2, 25/2, 45/2
To find the sum of the first ten terms of the AP, we can use the formula for the sum of an arithmetic series:
Sum = (n/2) * (2*a_1 + (n-1)*d)
Sum = (10/2) * (2*(5/2) + (10-1)*(5/2))
Sum = 5 * (5 + 9*(5/2))
Sum = 5 * (5 + 45/2)
Sum = 5 * (55/2)
Sum = 275/2
Therefore, the first four terms of the AP are 5/2, 15/2, 25/2, 45/2 and the sum of the first ten terms of the AP is 275/2.
Let's solve the problem step-by-step:
Step 1: Understanding the given information.
The second, fourth, and eighth terms of an arithmetic progression (AP) form the first three consecutive terms of a geometric progression (GP).
The sum of the third and fifth terms of the AP is 20.
Step 2: Define the terms.
Let's assume that the first term of the AP is "a" and the common difference is "d". Then, the terms of the AP can be written as:
1st term: a
2nd term: a + d
3rd term: a + 2d
4th term: a + 3d
5th term: a + 4d
...
n-th term: a + (n-1)d
Step 3: Find the terms of the GP.
Since the second, fourth, and eighth terms of the AP form the first three consecutive terms of the GP, we can write the terms of the GP as:
1st term: a + d
2nd term: a + 3d
3rd term: a + 7d
Step 4: Find the common ratio of the GP.
To find the common ratio of the GP, we can divide the second term by the first term:
Common ratio (r) = (a + 3d) / (a + d)
Step 5: Use the information to form an equation.
Since the sum of the third and fifth terms of the AP is 20, we can write the equation as:
(a + 2d) + (a + 4d) = 20
Step 6: Solve the equation.
Simplifying the equation, we get:
2a + 6d = 20
Divide both sides of the equation by 2:
a + 3d = 10
Step 7: Use the equation to find the values of a and d.
We have two equations:
Equation 1: r = (a + 3d) / (a + d)
Equation 2: a + 3d = 10
From Equation 2, we can express "a" in terms of "d":
a = 10 - 3d
Substituting this value of "a" in Equation 1, we get:
r = (10 - 3d + 3d) / (10 - 3d + d)
r = 10 / 10
r = 1
Step 8: Find the first four terms of the AP.
Using the equation for the n-th term of the AP (a + (n-1)d), we can calculate the values of the first four terms:
1st term (a) = 10 - 3d
2nd term (a + d) = 10 - 2d
3rd term (a + 2d) = 10 - d
4th term (a + 3d) = 10
So, the first four terms of the AP are (10 - 3d), (10 - 2d), (10 - d), and 10.
Step 9: Calculate the sum of the first ten terms of the AP.
The sum (S) of the first n terms of an AP can be calculated using the formula:
S = (n/2) * (2a + (n-1)d)
Since we want to find the sum of the first ten terms, we substitute n = 10:
S = (10/2) * (2(10 - 3d) + (10-1)d)
S = 5 * (20 - 6d + 9d)
S = 5 * (20 + 3d)
So, the sum of the first ten terms of the AP is 100 + 15d.