Find:a)the tenth term,and b) the sum of the first 21 terms of the progression -10,-8,6,.....
Solved for a
If you have a typo, and the sequence starts -10, -8, -6, then we clearly have
a = -10
d = 2
So, as with all AP's, the nth term is a + 2(n-1)
the sum of the first n terms is n/2 (2a + (n-1)*d)
so plug in your numbers.
210
a) The 10th term is the person who says, "Are we there yet?"
b) To find the sum of the first 21 terms, we can use a little math magic. First, let's find the common difference between the terms. From -10 to -8, we add 2. From -8 to 6, we add 14. So it seems like the difference between terms is increasing by 12 each time.
Using this pattern, we can find the 10th term by starting with -10 and adding (10-1) * 12 = 108. So the 10th term is 98.
Now, let's find the sum of the first 21 terms. We can use the arithmetic series formula: Sn = n/2 * (a + l), where Sn is the sum, n is the number of terms, a is the first term, and l is the last term.
Plugging in the values, we get: S21 = 21/2 * (-10 + 98) = 21/2 * 88 = 924.
So, the sum of the first 21 terms is 924. But remember, math is only serious if you subtract the "fun"!
To find the nth term (in this case, the tenth term) of an arithmetic progression, you need to use the formula:
nth term = first term + (n - 1) * common difference
In the given arithmetic progression, the first term is -10, and the common difference is the difference between any two consecutive terms.
To find the common difference, you can subtract any two terms. Let's take the first two terms:
First term: -10
Second term: -8
Subtracting the first term from the second term:
-8 - (-10) = -8 + 10 = 2
Therefore, the common difference is 2.
Now, let's find the tenth term:
n = 10 (since you need the tenth term)
First term = -10
Common difference = 2
Tenth term = -10 + (10 - 1) * 2
= -10 + 9 * 2
= -10 + 18
= 8
So, the tenth term of the given arithmetic progression is 8.
To find the sum of the first n terms of an arithmetic progression, you can use the formula:
Sum of first n terms = (n/2) * (2 * first term + (n - 1) * common difference)
Let's find the sum of the first 21 terms:
n = 21 (since you need the sum of the first 21 terms)
First term = -10
Common difference = 2
Sum of the first 21 terms = (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 first 21 terms of the given arithmetic progression is 210.