An arithmetic progression has 8terms.The sum of all 8terms is 180 and the sum of all odd terms is 80.

Calculate
(a)the first term and common difference
(b)the last term

ASAP..

I knw the (a) a=5,d=5
But the (b) how to do?

How to even do (a)?

To solve this problem, we can use the formulas for finding the sum of an arithmetic progression and the sum of an arithmetic series.

(a) To find the first term and common difference of the arithmetic progression, we can use the sum formulas.

The sum of all 8 terms in an arithmetic progression is given by the formula:

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

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

We are given that the sum of all 8 terms is 180, so we can substitute these values into the formula:

180 = (8/2)(2a + (8-1)d)
180 = 4(2a + 7d)
180 = 8a + 28d

Similarly, the sum of all odd terms in an arithmetic series is given by the formula:

S_odd = (n/2)(2a + (n-1)d),

where S_odd is the sum of the odd terms.

We are given that the sum of all odd terms is 80, so we can substitute these values into the formula:

80 = (4/2)(2a + (4-1)d)
80 = 2(2a + 3d)
80 = 4a + 6d

Now we have a system of equations with two unknowns (a and d):

Equation 1: 180 = 8a + 28d
Equation 2: 80 = 4a + 6d

We can solve this system of equations to find the values of a and d.

First, let's multiply Equation 2 by 2 to make the coefficients of a in both equations equal:

160 = 8a + 12d

Now we can subtract Equation 1 from this modified Equation 2:

160 - 180 = (8a + 12d) - (8a + 28d)
-20 = -16d
d = 1.25

Now we can substitute this value of d back into Equation 2 to find the value of a:

80 = 4a + 6(1.25)
80 = 4a + 7.5
4a = 72.5
a = 18.125

So, the first term (a) is 18.125 and the common difference (d) is 1.25.

(b) To find the last term of the arithmetic progression, we can use the formula:

an = a + (n-1)d

Substituting the values we found for a and d:

an = 18.125 + (8-1)(1.25)
an = 18.125 + 7 * 1.25
an = 18.125 + 8.75
an = 26.875

So, the last term of the arithmetic progression is 26.875.