in A.P the sum of the last five terms is 85 and the 4th is 9.what is the sum of the A.P?
d + d+k + d+2k + d+3k + d+4k
= 5d + 10k
4h term is d+3k = 9
5d + 10k = 85
5d + 15k = 45
-5k = 40
k = -5
and d = 9 + 15 = 24
If there are n terms, the sum of the last five is
a+(n-5)d + a+(n-2)d + a+(n-1)d
= 5a + (5n-15)d, so
a+(n-3)d = 17
a+3d=9
subtract to get
(n-6)d = 8
So, either
#1: n=7 d=8 a=-15
#1: n=8 d=4 a=-3
#3: n=10 d=2 a=3
#4: n=14 d=1 a=6
#1:
S7 = 7/2(-30+48) = 63
S2 = 2/2(-30+8) = -24
sum of last 5 is S7-S2 = 85
#2:
S8 = 8/2(-6+28) = 88
S3 = 3/2(-6+8) = 3
sum of last 5 is S8-S3 = 85
#3:
S10 = 10/2(6+18) = 120
S5 = 5/2(6+8) = 35
sum of last 5 is S10-S5 = 85
#4:
S14 = 14/2(12+13) = 175
S9 = 9/2(12+8) = 90
sum of last 5 is S14-S9 = 85
Sounds like the problem was poorly worded.
To find the sum of an arithmetic progression (A.P.), you need to know three things: the first term (a), the common difference (d), and the number of terms (n).
Given:
Sum of the last five terms = 85
Fourth term (a4) = 9
To find the sum of the A.P., we can follow these steps:
Step 1: Find the common difference (d):
Since a4 = a + 3d (the fourth term is the first term plus 3 times the common difference in an A.P.), we can substitute the known values:
9 = a + 3d
Step 2: Solve for the first term (a):
Rearrange the equation derived in Step 1 to solve for a:
a = 9 - 3d
Step 3: Find the last term (an):
In an A.P., the last term (an) is given by:
an = a + (n-1)d
Since the sum of the last five terms is 85, we can form the equation:
85 = 5/2 * [2a + (5-1)d]
Step 4: Rearrange the equation from Step 3 to solve for n:
Simplify the equation:
85 = 5/2 * [2a + 4d]
Multiply both sides by 2/5 to get:
34 = 2a + 4d
Substitute the value of 'a' derived in Step 2:
34 = 2(9 - 3d) + 4d
34 = 18 - 6d + 4d
34 - 18 = -2d
d = -16/(-2)
d = 8
Step 5: Solve for the first term (a):
Substitute the value of 'd' derived in Step 4 back into the equation from Step 2:
a = 9 - 3(8)
a = 9 - 24
a = -15
Step 6: Calculate the number of terms (n):
Using the given sum of the last five terms:
85 = 5/2 * [2(-15) + (5-1)(8)]
85 = 5/2 * [-30 + 4(8)]
85 = 5/2 * [-30 + 32]
85 = 5/2 * 2
85 = 5(1)
n = 5
Step 7: Calculate the sum of the A.P. using the formula:
Sum = n/2 * [2a + (n-1)d]
Substitute the values of 'a', 'd', and 'n' derived in previous steps:
Sum = 5/2 * [2(-15) + (5-1)(8)]
Sum = 5/2 * [-30 + 4(8)]
Sum = 5/2 * [-30 + 32]
Sum = 5/2 * 2
Sum = 5(1)
Sum = 5
Therefore, the sum of the arithmetic progression (A.P.) is 5.