In an AP whose first term is 2,the sum of first five term is one fourth the sum of the next five terms.show that T20=-112. Find S20.
we have
a=2
sum of 1st 5 terms: S5
sum of next 5 terms: S10-S5
So, we now have
S5 = (S10-S5)/4
4S5 = S10-S5
5S5 = S10
(5)(5/2)(4+4d) = (10/2)(4+9d)
d = -6
The progression is thus
2, -4, -10, -16, -22, -28, -34, -40, -46, -52 ...
S5 = -50
S10 = -250
S10-S5 = -200
T20 = 2+19(-6) = -112
S20 = (20/2)(4+19(-6)) = -1100
To solve this problem, we first need to find the common difference (d) of the arithmetic progression (AP). Given that the sum of the first five terms (S5) is one-fourth the sum of the next five terms (S10), we can set up the following equation:
S5 = (1/4) * S10
Using the formula for the sum of an arithmetic series, which is given by:
Sn = (n/2) * (2a + (n-1)d)
where Sn represents the sum of the first 'n' terms, 'a' is the first term, and 'd' is the common difference, we can calculate S5 and S10 as follows:
S5 = (5/2) * (2 + (5-1)d)
S10 = (10/2) * (2 + (10-1)d)
Substituting these values into the equation from before, we get:
(5/2) * (2 + (5-1)d) = (1/4) * (10/2) * (2 + (10-1)d)
Simplifying and canceling out common terms:
5 * (2 + 4d) = 1/2 * (2 + 9d)
10 + 20d = 1 + 4.5d
Multiplying through by 2 to remove fractions:
20 + 40d = 2 + 9d
Subtracting 9d from both sides:
20 + 31d = 2
Subtracting 20 from both sides:
31d = -18
Dividing by 31:
d = -18/31
Now that we have the common difference (d), we can find the 20th term (T20) using the formula:
Tn = a + (n-1)d
where Tn represents the nth term. Since we know the first term 'a' is 2, we can substitute these values into the formula to find T20:
T20 = 2 + (20-1) * (-18/31)
Simplifying:
T20 = 2 + 19 * (-18/31)
T20 = 2 - (18*19)/(31)
T20 = -112/31
Therefore, T20 = -112/31.
To find the sum of the first 20 terms (S20), we can again use the formula for the sum of an arithmetic series:
S20 = (20/2) * (2 + (20-1) * (-18/31))
Simplifying:
S20 = 10 * (2 + 19 * (-18/31))
S20 = 10 * (2 - 342/31)
S20 = 10 * (2 * 31/31 - 342/31)
S20 = 10 * (62/31 - 342/31)
S20 = 10 * (-280/31)
S20 = -2800/31
Therefore, S20 = -2800/31.