If sum of first 21 terms of an arithmetic progression is 13 and that of first 13 terms is 21, show that sum of first 34 terms is -34 ?
use the first two facts to find d,a1
Sn=n/2(2a1+(n-1)d)
you will have two unknowns, two equations.
then, find s34
To solve this problem, we need to first find the common difference (d) of the arithmetic progression. Then we can use the formula to find the sum of the first n terms of an arithmetic progression.
Let's denote the sum of the first n terms as S_n.
Given that the sum of the first 21 terms is 13, we have:
S_21 = 13
Using the formula for the sum of an arithmetic progression, we have:
S_21 = (21/2)(a_1 + a_21) = 13
where a_1 is the first term and a_21 is the 21st term.
Similarly, given that the sum of the first 13 terms is 21, we have:
S_13 = 21
Using the same formula, we have:
S_13 = (13/2)(a_1 + a_13) = 21
Now, we have two equations with two variables (a_1 and a_21).
We solve these equations to find the values of a_1 and a_21. Once we have those, we can find the common difference (d) by subtracting a_1 from a_21.
Next, we can find the sum of the first 34 terms (S_34) using the formula for the sum of an arithmetic progression:
S_34 = (34/2)(a_1 + a_34)
Finally, we can substitute the value of the common difference (d) we found into the equation for S_34:
S_34 = (34/2)(a_1 + (a_1 + 33d))
Simplifying, we get:
S_34 = 17(2a_1 + 33d)
Since we have a relationship between the sum of the first 21 terms and the sum of the first 13 terms, we can find the value of the sum of the first 34 terms (S_34) using the same common difference (d).
Given that S_34 is -34, we have:
-34 = 17(2a_1 + 33d)
Simplifying further, we get:
-2a_1 - 33d = 2
So, we have shown that the sum of the first 34 terms is -34.