Given that the sum of the first 20 terms of an arithmetic sequence is 1150 and the fifth term is twice the second term .

find:

1.The first term and the common difference
2. find the general formula

term5 = a+4d

term2 = a+d

a+4d = 2(a+d)
a+4d = 2a + 2d
a = 2d

sum(20) = 10(2a + 19d) = 1150
2a + 19d = 115
4d+19d=115
23d = 115
d = 5
a = 10

you MUST know how to do this simple kind of question.
Repeat the solution on your own, without looking at mine. Practise, practise, practise ....

To find the first term and the common difference of an arithmetic sequence, we will use the given information.

Let's denote the first term as 'a' and the common difference as 'd'.

We know that the sum of the first 20 terms is 1150, so we can use the formula for the sum of an arithmetic series:

Sum = (n/2) * (2a + (n-1)d)

Here, 'n' represents the number of terms. In our case, n = 20.

1. Find the first term and common difference:
Given: Sum = 1150, n = 20

Plugging in the values into the formula:
1150 = (20/2) * (2a + (20-1)d)
1150 = 10 * (2a + 19d)
115 = 2a + 19d ............(Equation 1)

We also know that the fifth term is twice the second term:
a + 4d = 2(a + d)
a + 4d = 2a + 2d
2d - d = a
d = a

Now, substitute d = a in equation 1:
115 = 2a + 19(a)
115 = 2a + 19a
115 = 21a
a = 115/21
a = 5.48 (rounded to two decimal places)
So, the first term (a) is approximately 5.48.

Since we found that the common difference (d) is equal to the first term (a), the common difference in this arithmetic sequence is also approximately 5.48.

Therefore, the first term is approximately 5.48, and the common difference is also approximately 5.48.

2. Find the general formula:
The general formula for an arithmetic sequence is given by:
An = a + (n-1)d

Using the values we found:
An = 5.48 + (n - 1) * 5.48
or,
An = 5.48n

So, the general formula for this arithmetic sequence is An = 5.48n.

To find the first term (a) and the common difference (d) of an arithmetic sequence, we can use the following steps:

1. Use the formula for the sum of the first n terms of an arithmetic sequence: Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.

2. Plug in the given values into the formula for Sn: 1150 = (20/2)(2a + (20-1)d).

3. Simplify the equation: 1150 = 10(2a + 19d).

4. Divide both sides of the equation by 10: 115 = 2a + 19d.

5. Rearrange the equation to solve for a: a = (115 - 19d)/2.

6. Use the given information that the fifth term is twice the second term: a + 4d = 2(a + d).

7. Simplify the equation: a + 4d = 2a + 2d.

8. Rearrange the equation to solve for a: a = 2d.

9. Substitute the value of a from step 8 into the equation from step 5: 2d = (115 - 19d)/2.

10. Multiply both sides of the equation by 2: 4d = 115 - 19d.

11. Add 19d to both sides of the equation: 23d = 115.

12. Divide both sides of the equation by 23: d = 5.

13. Substitute the value of d into the equation from step 8 to find a: a = 2(5) = 10.

Therefore, the first term (a) is 10 and the common difference (d) is 5.

To find the general formula for the arithmetic sequence, we can use the formula: nth term (An) = a + (n-1)d, where An is the nth term.

In this case, the first term (a) is 10 and the common difference (d) is 5.

So, the general formula for the arithmetic sequence is: An = 10 + (n-1)5.