The 10th and 15th term of an Ap are -5 and -7½ respectively, what is the sum of the first 20 term
You must learn the main definitions of sequences, since you will need them
10th term is -5 ----> a + 9d = -5
15th term is -7.5 ---> a + 14d = -7.5
subtract them:
5d = -2.5
d = -.5
then in a + 14d = -7.5
a - 7 = -7.5
a = -.5
now you have a and d, using sum(n) = (n/2)(2a + (n-1)d )
sum(20) = (20/2)(2(-.5) + 19(-.5) )
= ....
let me know what you get
-105
shoe full work
40
Why did the math teacher open a bakery? Because she wanted to knead and multiply dough! Now, let's solve this arithmetic progression problem, shall we?
To find the first term (a) and the common difference (d), we can use the formula for the nth term of an arithmetic progression:
a_n = a + (n - 1)d
Given that the 10th term is -5 and the 15th term is -7½, we can substitute these values into the formula. Let's do some math magic!
For the 10th term:
-5 = a + (10 - 1)d
For the 15th term:
-7.5 = a + (15 - 1)d
Now, let's solve these equations simultaneously. One option is to subtract the first equation from the second equation:
-7.5 - (-5) = (a + (15 - 1)d) - (a + (10 - 1)d)
Simplifying this, we have:
-2.5 = 4d
Dividing both sides by 4:
d = -2.5/4
d = -0.625
Now that we know the common difference (d), we can use it to find the first term (a). Using the first equation:
-5 = a + (10 - 1)(-0.625)
Simplifying this equation, we get:
-5 = a - 6.25
Adding 6.25 to both sides:
1.25 = a
Now we have found the first term (a). Great work!
To find the sum of the first 20 terms, we can use the formula for the sum of an arithmetic progression:
S_n = (n/2)(2a + (n - 1)d)
Substituting the values into the formula:
S_20 = (20/2)(2(1.25) + (20 - 1)(-0.625))
Simplifying this expression, we have:
S_20 = 10(2.5 + 19(-0.625))
S_20 = 10(2.5 - 11.875)
S_20 = 10(-9.375)
S_20 = -93.75
So, the sum of the first 20 terms of the arithmetic progression is -93.75. Keep up the great work, my mathematical friend!
To find the sum of the first 20 terms of the arithmetic progression, we first need to find the common difference between each term.
The common difference (d) can be calculated using the formula:
d = (n_th term - (n-1)_th term)
Given that the 10th term is -5 and the 15th term is -7½, we can calculate the common difference as follows:
d = (-7½ - (-5)) = (-7½ + 5) = -2½
Now that we have the common difference, we can find the sum of the first 20 terms of the arithmetic progression using the formula for the sum of an arithmetic series:
S_n = (n/2) * (2a + (n-1)d)
Where:
S_n = sum of the first n terms
a = first term
n = number of terms
d = common difference
In this case, we want to find the sum of the first 20 terms. We know that the common difference (d) is -2½, and we need to find the first term (a).
To determine the first term, we can use the formula for the n_th term of an arithmetic progression:
a_n = a + (n-1)d
We know that the 10th term (a_10) is -5. Plugging this information into the formula, we can solve for the first term (a):
-5 = a + (10-1)(-2½)
-5 = a + 9(-2½)
-5 = a + (-22½)
-5 + 22½ = a
17½ = a
Now that we have the first term (a) and the common difference (d), we can find the sum of the first 20 terms (S_20) using the formula:
S_20 = (20/2) * (2 * 17½ + (20-1) * -2½)
Simplifying this expression:
S_20 = 10 * (35 + 19 * -2½)
S_20 = 10 * (35 + (-47½))
S_20 = 10 * (35 - 47½)
S_20 = 10 * (-12½)
S_20 = -125
Therefore, the sum of the first 20 terms of the arithmetic progression is -125.