The 10th and 15th terms of an AP are -5 and 7 1/2 respectively.What is the sum of the first 20 terms
look at the post 2 above this how I found the a and d
Then use your sum formula to find sum(20)
To find the sum of the first 20 terms of an arithmetic progression (AP), we need to use the formula for the sum of an AP. The formula is:
Sn = (n/2)(a + l)
Where Sn is the sum of the first n terms, a is the first term, l is the last term, and n is the number of terms.
In this case, we are given that the 10th term is -5 and the 15th term is 7 1/2. To find the first term (a) and the last term (l), we can use the formula for the nth term of an AP:
an = a + (n-1)d
Where an is the nth term, a is the first term, n is the term number, and d is the common difference.
From the given information, we can calculate the common difference (d) by subtracting the 10th term from the 15th term:
d = 7 1/2 - (-5) = 7 1/2 + 5 = 12 1/2 = 25/2
Now, we can substitute the values into the formula to find the first term (a):
-5 = a + (10 - 1)(25/2)
Simplifying the equation:
-5 = a + 9(25/2)
-5 = a + 225/2
-5 = (2a + 225)/2
-5 * 2 = 2a + 225
-10 - 225 = 2a
-235 = 2a
a = -235/2
Therefore, the first term (a) is -235/2 and the common difference is 25/2.
Now, we have all the information we need to find the sum of the first 20 terms (Sn):
Sn = (n/2)(a + l)
S20 = (20/2)(-235/2 + l)
To find the last term (l), we can use the formula for the nth term of an AP:
l = a + (n-1)d
l = -235/2 + (20-1)(25/2)
l = -235/2 + 19(25/2)
Now, substitute the values into the formula for the sum:
S20 = (20/2)(-235/2 + -235/2 + 19(25/2))
Simplifying the equation:
S20 = (10)(-235 + -235 + 475)
S20 = (10)(-470 + 475)
S20 = (10)(5)
S20 = 50
Therefore, the sum of the first 20 terms is 50.
Given that the 10th term (a10) of the arithmetic progression (AP) is -5, and the 15th term (a15) is 7 1/2, we need to find the sum of the first 20 terms of the AP.
Step 1: Find the common difference (d)
The common difference (d) can be calculated using the formula:
d = (a15 - a10) / (15 - 10)
= (7 1/2 - (-5)) / 5
= (15/2 + 5) / 5
= (15/2 + 10/2) / 5
= 25/2 / 5
= 25/10
= 5/2
= 2.5
Therefore, the common difference (d) is 2.5.
Step 2: Find the first term (a1) of the AP
We can find the first term (a1) by substituting any of the given terms in the formula: a1 = a10 - (10 - 1) * d
Let's use a10 = -5:
a1 = -5 - (10 - 1) * 2.5
= -5 - 9 * 2.5
= -5 - 22.5
= -27.5
Therefore, the first term (a1) of the AP is -27.5.
Step 3: Find the sum of the first 20 terms (S20) of the AP
The sum of the first 20 terms (S20) of an AP can be calculated using the formula:
S20 = (n/2) * (a1 + an)
where n is the number of terms, a1 is the first term, and an is the last term.
We already know a1 = -27.5. We need to find the 20th term (a20) of the AP to calculate the sum.
Step 4: Find the 20th term (a20) of the AP
a20 = a1 + (20 - 1) * d
= -27.5 + (19) * 2.5
= -27.5 + 47.5
= 20
Therefore, the 20th term (a20) of the AP is 20.
Step 5: Calculate the sum of the first 20 terms (S20)
S20 = (n/2) * (a1 + an)
= (20/2) * (-27.5 + 20)
= 10 * (-7.5)
= -75
Therefore, the sum of the first 20 terms of the AP is -75.