The 6th term of an arithmetical progression is 20 and the 11th term is -5. find the sum of the first three term please bot solve in details
Let's denote the first term of the arithmetic progression as "a" and the common difference as "d".
The formula to find the nth term of an arithmetic progression is:
nth term = a + (n-1)d
Given that the 6th term is 20, we have:
6th term = a + (6-1)d = a + 5d = 20 -- equation (1)
Given that the 11th term is -5, we have:
11th term = a + (11-1)d = a + 10d = -5 -- equation (2)
To find the sum of the first three terms, we need to find the values of "a" and "d" first.
We can solve the above two equations simultaneously to find "a" and "d". Subtracting equation (1) from equation (2), we get:
(a + 10d) - (a + 5d) = -5 - 20
5d = -25
d = -5
Substituting the value of "d" into equation (1), we can find "a":
a + 5(-5) = 20
a - 25 = 20
a = 45
Now that we have "a" and "d", we can find the sum of the first three terms:
Sum of first three terms = 3a + 3(a+d) + 3(a+2d)
= 3(45) + 3(45+(-5)) + 3(45+2(-5))
= 135 + 135 + 135
= 405
Please bot explain to me very well the two last equation you solved I don't understand it please explain
Certainly! I apologize for any confusion.
To find the sum of the first three terms, we can use the formula:
Sum of first three terms = a + (a + d) + (a + 2d)
Here, "a" represents the first term, and "d" is the common difference that we found earlier.
Substituting the values of "a" and "d" into the equation, we get:
Sum of first three terms = 45 + (45 + (-5)) + (45 + 2(-5))
Let's break down the equation step by step:
1. First, we substitute the value of "a" (which is 45) into the equation:
Sum of first three terms = 45 + (___ + (-5)) + (___ + 2(-5))
2. Next, we substitute the values of "a" and "d" into the equation to fill in the blanks:
Sum of first three terms = 45 + (45 + (-5)) + (45 + 2(-5))
= 45 + 45 + (-5) + 45 + 2(-5)
Note: In this step, when we multiply "2" with "(-5)", we get "-10".
3. We can simplify the addition and subtraction:
Sum of first three terms = 45 + 45 + (-5) + 45 + (-10)
4. Combining the like terms, we get:
Sum of first three terms = 135 + (-5) + (-10)
5. Adding the numbers together, we obtain the final result:
Sum of first three terms = 135 - 5 - 10
= 120
Therefore, the sum of the first three terms is 120.