find:
a)the tenth term
b)the sum of the 21st terms of the progression:-10,-8,-6
a = -10
d = 2
(a) a10 = a+9d
(b) S21 = 21/2 (2a+20d)
To find the tenth term of the progression -10, -8, -6, we need to determine the pattern of the sequence.
The given sequence is an arithmetic progression with a common difference of 2. This means that each term in the sequence is obtained by adding 2 to the previous term.
The first term in the sequence is -10, the second term is -10 + 2 = -8, and the third term is -10 + 2(2) = -6.
We can continue this pattern to find the tenth term.
The fourth term will be -6 + 2 = -4, the fifth term will be -4 + 2 = -2, and so on.
To find the tenth term, we can use the formula:
nth term = first term + (n-1) * common difference
In this case, the first term is -10, n is 10, and the common difference is 2.
So, the tenth term is:
-10 + (10-1)*2 = -10 + 9*2 = -10 + 18 = 8
Therefore, the tenth term of the progression -10, -8, -6 is 8.
b) To find the sum of the 21st terms in the progression -10, -8, -6, we can use the formula for the sum of an arithmetic series:
S = n/2 * (2a + (n - 1) * d)
where S is the sum of the terms, n is the number of terms, a is the first term, and d is the common difference.
In this case, we want to find the sum of the first 21 terms, so n = 21, a = -10, and d = 2.
Plugging in these values into the formula, we get:
S = 21/2 * (2*(-10) + (21 - 1) * 2)
= 21/2 * (-20 + 20*2)
= 21/2 * (-20 + 40)
= 21/2 * 20
= 21 * 10
= 210
Therefore, the sum of the 21st terms of the progression -10, -8, -6 is 210.
To find the tenth term of the given arithmetic progression and the sum of the 21st terms, we need to determine the common difference (d).
In an arithmetic progression, each term is obtained by adding a constant difference (d) to the previous term.
Given the progression -10, -8, -6, we can observe that each term is obtained by adding 2 to the previous term. Therefore, the common difference (d) is 2.
Now, we can use this information to find the required values:
a) To find the tenth term, we can use the formula for the nth term of an arithmetic progression:
nth term (Tn) = a + (n-1)d,
where Tn is the nth term, a is the first term, n is the position of the term, and d is the common difference.
In our case, a = -10 (the first term), n = 10 (the position), and d = 2 (the common difference).
Plugging these values into the formula, we get:
T10 = -10 + (10-1) * 2
= -10 + 9 * 2
= -10 + 18
= 8.
Therefore, the tenth term of the given arithmetic progression is 8.
b) To find the sum of the 21st terms, we can use the formula for the sum of an arithmetic progression:
Sn = (n/2) * (2a + (n-1)d),
where Sn is the sum of the first n terms, a is the first term, n is the number of terms, and d is the common difference.
In our case, a = -10 (the first term), n = 21 (the number of terms), and d = 2 (the common difference).
Plugging these values into the formula, we get:
S21 = (21/2) * (2*(-10) + (21-1)*2)
= (21/2) * (2*(-10) + 20*2)
= (21/2) * (-20 + 40)
= (21/2) * 20
= 21 * 10
= 210.
Therefore, the sum of the 21st terms in the given arithmetic progression is 210.