1) The first term of arithmetic progression is -20 and the sum of it's term is 250. find it's last term if the number of it's terms is 10

2) Find the fifth term from the arithmetic progression -12, -9, -6 hence find the sum of it's first fifty terms.

3) The sum of the first terms in Arithmetic progression is equal to -24, and the sum of the last six terms equal to 30. if the number of it's term is 12. find a) it's common different
b) The first term
c) the last term

Answer

3)a=-12

d=3

t(5)=a+(n-1)d
=-12+(5-1)3
=0

s(50)=3625

hope it would be a help. :)

U2= a=8/9 U2=8/9 Un= arn-1 if n=2 U2= arn7-1 8/9=2(r6)

1) a=-20

Sn=250
. n=10
Sn=n/2(2a+(n-1)d)
250=10/2(2(-20)+(10-1)d)
250=5((-40)+(9)d)
250=-200+45d
250+200=45d
d=10
Un=a+(n-1)d
Un=-20*(10-1)10
Un=-20+900
Un=880
::the last term=880

2) a=-12

. Un=-6
d=-3
n=5

S5=5/2(2(-12)+(5-1)-3)
2.5(-24)+-12)
. 2.5(-36)
S5=-90

1) Sn= n/2 (a + l )

250=10/2 (-20+l)
250=(-200+10l)/2
250 x 2= -200+10L
500=-200+10L
500+200= 10L
700=10L

L= 700/10
L=70

1) To find the last term of an arithmetic progression, we need to know the first term, the number of terms, and the sum of all the terms.

Given:
First term (a) = -20
Number of terms (n) = 10
Sum of terms (S) = 250

We can use the formula for the sum of an arithmetic progression:

S = (n/2) * (2a + (n - 1)d)

where S is the sum, n is the number of terms, a is the first term, and d is the common difference.

Substituting the given values:

250 = (10/2) * (2 * -20 + (10 - 1)d)
250 = 5 * (-40 + 9d)
50 = -40 + 9d
90 = 9d
d = 10

Now that we have found the common difference, we can find the last term using the formula:

last term = first term + (number of terms - 1) * common difference
last term = -20 + (10 - 1) * 10
last term = -20 + 9 * 10
last term = -20 + 90
last term = 70

Therefore, the last term of the arithmetic progression is 70.

2) To find the fifth term of an arithmetic progression, we need to know the first term and the common difference.

Given:
First term (a) = -12
Common difference (d) = -3

To find the fifth term, we can use the formula:

nth term = first term + (n - 1) * common difference

Substituting the given values:

fifth term = -12 + (5 - 1) * (-3)
fifth term = -12 + 4 * (-3)
fifth term = -12 + (-12)
fifth term = -24

Therefore, the fifth term of the arithmetic progression is -24.

To find the sum of the first fifty terms, we can use the formula for the sum of an arithmetic progression:

Sum = (n/2) * (2a + (n - 1)d)

Given:
First term (a) = -12
Number of terms (n) = 50
Common difference (d) = -3

Substituting the given values:

Sum = (50/2) * (2 * -12 + (50 - 1) * -3)
Sum = 25 * (-24 + 49 * -3)
Sum = 25 * (-24 - 147)
Sum = 25 * (-171)
Sum = -4275

Therefore, the sum of the first fifty terms of the arithmetic progression is -4275.

3) Given:
Sum of the first terms (S1) = -24
Sum of the last six terms (S2) = 30
Number of terms (n) = 12

We need to find:
a) Common difference (d)
b) First term (a)
c) Last term (L)

To find the common difference, we can use the formula for the sum of an arithmetic progression:

S = (n/2) * (2a + (n - 1)d)

Since we know the sum of the first and last six terms, we can subtract S2 from S1 to find the sum of the first six terms (S3) and the sum of the remaining terms (S4):

S3 = S1 - S2
S3 = -24 - 30
S3 = -54

S4 = S1 - S3
S4 = -24 - (-54)
S4 = -24 + 54
S4 = 30

Now, we can use S3 and S4 to find the common difference:

S3 = (6/2) * (2a + (6 - 1)d)
-54 = 3 * (2a + 5d)

S4 = (6/2) * (2a + (6 - 1)d)
30 = 3 * (2a + 5d)

Dividing the two equations:

-54/30 = (2a + 5d)/(2a + 5d)
-54/30 = 1

This means the common difference (d) cancels out, which provides no information. Therefore, we cannot determine the common difference from the given information.

To find the first term (a), we can substitute the values into one of the equations:

-54 = 3 * (2a + 5d)

Simplifying the equation:

-54 = 6a + 15d

Substituting d = -2a in the equation:

-54 = 6a + 15(-2a)
-54 = 6a - 30a
-24a = -54
a = 2.25

Therefore, the first term of the arithmetic progression is 2.25.

To find the last term (L), we can use the formula:

L = a + (n - 1) * d

Substituting the values:

L = 2.25 + (12 - 1) * d
L = 2.25 + 11d

Since we don't know the common difference (d), we cannot determine the exact value of the last term (L) without additional information.