In an arithmetic sequence the sum of the first ten terms is 400 and the sum of next ten terms is 1000. Find

The common difference


The first term


The first three terms of arithmetic sequence are also 2x, x+4 and 2x-7. Verify that when x = 8 the term form a geometric sequence and find the sum to infinity in this case.
S

To find the common difference and the first term in the arithmetic sequence, we can use the formulas for the sum of an arithmetic sequence.

1. Finding the common difference:

Let's denote the common difference as "d."

The sum of the first ten terms of an arithmetic sequence can be given by the formula:

S₁ = (n/2)(2a + (n-1)d)

Where S₁ is the sum of the first n terms, a is the first term, and d is the common difference.

In this case, S₁ is 400 and n is 10.

400 = (10/2)(2a + 9d)
400 = 5(2a + 9d)
80 = 2a + 9d.......(Eq. 1)

Similarly, the sum of the next ten terms can be given by the formula:

S₂ = (n/2)(2a + (n-1)d)

Where S₂ is the sum of the next n terms.

In this case, S₂ is 1000 and n is 10.

1000 = (10/2)(2a + 9d)
1000 = 5(2a + 9d)
200 = 2a + 9d.......(Eq. 2)

Now, we have two equations (Eq. 1 and Eq. 2) with two unknowns (a and d). We can use these equations to solve for a and d.

Let's subtract Eq. 2 from Eq. 1:

80 - 200 = (2a + 9d) - (2a + 9d)
-120 = 0

Since we obtained an equality that is not true, there is no solution for a and d that satisfies both equations. This means that there is an error in the given information.

2. Verification of a geometric sequence:

To verify if the terms form a geometric sequence, we need to check if the ratio between consecutive terms is constant.

Given the first three terms of the arithmetic sequence as 2x, x+4, and 2x-7, we can find the ratio between consecutive terms:

Ratio between second and first term:
(x+4) / (2x) = (1/2)

Ratio between third and second term:
(2x-7) / (x+4) = (2x-7) / (x+4)

For x = 8:

Ratio between second and first term:
((8+4) / (2*8)) = (12 / 16) = (3/4)

Ratio between third and second term:
((2*8-7) / (8+4)) = (9 / 12) = (3/4)

Since the ratio between consecutive terms is constant (3/4), the terms form a geometric sequence when x = 8.

3. Finding the sum to infinity in the geometric sequence:

To find the sum to infinity of a geometric sequence, we use the formula:

S = a / (1 - r)

Where S is the sum to infinity, a is the first term, and r is the common ratio.

In this case, the first term of the geometric sequence is 2x, and the ratio is (3/4).

For x = 8:

a = 2*8 = 16
r = 3/4

S = 16 / (1 - (3/4))
S = 16 / (1/4)
S = 16 * 4
S = 64

Therefore, when x = 8, the terms form a geometric sequence and the sum to infinity is 64.

To find the common difference in an arithmetic sequence, we can use the formula for the sum of the terms in an arithmetic sequence. Let's call the common difference "d" and the first term "a". The sum of the first ten terms is given as 400, so we have:

S_10 = (n/2)(2a + (n-1)d) = 400

where n is the number of terms in the sum, which in this case is 10. We can substitute n=10 into the equation:

400 = (10/2)(2a + (10-1)d)
400 = 5(2a + 9d)

Similarly, for the sum of the next ten terms, we have:

S_20 = (n/2)(2a + (n-1)d) = 1000

where n is again 10:

1000 = (10/2)(2a + (10-1)d)
1000 = 5(2a + 9d)

Now we have a system of two equations with two variables (a and d):

400 = 5(2a + 9d)
1000 = 5(2a + 9d)

We can solve these equations simultaneously to find the values of a and d.

Now let's verify if the first three terms of the arithmetic sequence 2x, x+4, and 2x-7 form a geometric sequence when x = 8. In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio. Let's calculate the ratios:

(x+4) / (2x) = (2x-7) / (x+4)

Simplifying these ratios gives:

(x+4)^2 = (2x)(2x-7)

Expanding and simplifying further gives:

x^2 + 8x + 16 = 4x^2 - 14x

Rearranging the equation and simplifying gives:

3x^2 - 22x + 16 = 0

To find the values of x that make this equation true, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a=3, b=-22, and c=16. Plugging in these values and solving the equation gives two possible values for x. We need to check both values to see if they make the terms form a geometric sequence.

Once we find the correct value(s) of x, we can check if the terms form a geometric sequence by calculating the ratio between consecutive terms. If the ratio is the same for all consecutive terms, we have a geometric sequence.

To find the sum of an infinite geometric sequence, we can use the formula:

S = a / (1 - r)

where a is the first term and r is the common ratio. In this case, we can substitute the appropriate values of a and r to find the sum to infinity.

#1. Using your usual formulas, you have

10/2 (2a+9d) = 400
20/2 (2a+19d) - 400 = 1000
Now solve for a and d

#2. You know that
(x+4)-2x = (2x-7)-(x+4)
x = 15/2
If x=8, the three terms are 16,12,9
So S = a/(1-r) = 16/(1 - 3/4) = 64