The 10th and 15th term of an arithmetic progression are -5 and minus 7 whole number 1 over 2 what is the sum of the 1st 20 term
To find the sum of the first 20 terms of an arithmetic progression, we need to know the first term and the common difference.
Let's say the first term is a and the common difference is d.
We are given that the 10th term is -5 and the 15th term is -7 1/2.
Using the formula for the nth term of an arithmetic progression:
nth term = a + (n-1)d
We can set up two equations using the given information:
-5 = a + (10-1)d
-7 1/2 = a + (15-1)d
Simplifying the equations:
-5 = a + 9d
-7 1/2 = a + 14d
We can multiply the second equation by 2 to get rid of the fraction:
-15 = 2a + 28d
Now we have a system of three equations:
-5 = a + 9d
-7 1/2 = a + 14d
-15 = 2a + 28d
Solving this system of equations, we find that a = -57/2 and d = 5/2.
Now we have the first term a = -57/2 and the common difference d = 5/2.
The sum of the first 20 terms can be calculated using the formula for the sum of an arithmetic progression:
Sum = (n/2)(2a + (n-1)d)
Substituting n = 20, a = -57/2, and d = 5/2:
Sum = (20/2)(2(-57/2) + (20-1)(5/2))
= 10(-57 + 19(5/2))
= 10(-57 + 95/2)
= 10(-57 + 47.5)
= 10(-9.5)
= -95
Therefore, the sum of the first 20 terms of the arithmetic progression is -95.
To find the sum of the first 20 terms of an arithmetic progression, we first need to find the common difference (d) of the sequence.
Given that the 10th term is -5 and the 15th term is -7 1/2, we can set up the following equations:
a + 9d = -5 ---- (1) (The 10th term is the first term plus 9 times the common difference)
a + 14d = -7 1/2 ---- (2) (The 15th term is the first term plus 14 times the common difference)
To solve these equations, we can subtract equation (1) from equation (2) to eliminate the "a" term:
(a + 14d) - (a + 9d) = -7 1/2 - (-5)
5d = -7 1/2 + 5
5d = -2 1/2
5d = -5/2
d = -1/2
Now that we have the common difference (d = -1/2), we can find the first term (a) using equation (1):
a + 9(-1/2) = -5
a - 9/2 = -5
a = -5 + 9/2
a = -5 + 4 1/2
a = -1/2
We have found that the first term (a) is -1/2 and the common difference (d) is -1/2. Now we can calculate the sum of the first 20 terms of the arithmetic progression using the formula:
S = (n/2) * (2a + (n-1) * d)
where S is the sum, n is the number of terms (20 in this case), a is the first term, and d is the common difference.
Substituting the values we know into the formula:
S = (20/2) * (2 * -1/2 + (20-1) * -1/2)
S = 10 * (-1 + 19 * -1/2)
S = 10 * (-1 - 19/2)
S = 10 * (-2/2 - 19/2)
S = 10 * (-21/2)
S = -210/2
S = -105
Therefore, the sum of the first 20 terms of the arithmetic progression is -105.
To find the sum of the first 20 terms of an arithmetic progression, we need to find the common difference (d) and the first term (a1) of the sequence. With the given information, we can calculate these values as follows:
Given:
a10 = -5
a15 = -7 1/2
Step 1: Finding the Common Difference (d)
To find the common difference (d), we can use the formula:
d = a15 - a10
Substituting the given values:
d = (-7 1/2) - (-5)
d = -7 1/2 + 5
To add these two fractions, we need to find a common denominator, which is 2. So:
d = (-7/2) + (10/2)
d = (3/2)
Therefore, the common difference (d) is 3/2.
Step 2: Finding the First Term (a1)
To find the first term (a1), we can use the formula:
a1 = a10 - 9d
Substituting the given values:
a1 = (-5) - 9(d)
a1 = (-5) - 9(3/2)
a1 = (-5) - (27/2)
To subtract fractions, we need a common denominator of 2:
a1 = (-10/2) - (27/2)
a1 = (-37/2)
Therefore, the first term (a1) is -37/2.
Step 3: Finding the Sum of the First 20 Terms
To find the sum of the first 20 terms, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(2a1 + (n-1)d)
Substituting the values:
n = 20 (as we want to find the sum of first 20 terms)
a1 = -37/2
d = 3/2
Sn = (20/2)(2(-37/2) + (20-1)(3/2))
Sn = 10(-37 + 19(3/2))
To simplify the expression, we need to find the common denominator (2):
Sn = 10(-(37/2) + (57/2))
Sn = 10(20/2)
Sn = 10(10)
The sum of the first 20 terms is 100.