Find the number of terms and the sum of the terms of the arithmetic progressions 32,28,….4a.14 : 252b.8 : 144c.15 : 270d.7 : 126

Sure, let's solve this step by step:

a. 32, 28, …, 4

The common difference (d) here is 28 - 32 = -4. The first term (a) is 32 and the last term (l) is 4.

The number of terms (n) in an arithmetic progression can be found using the formula: n = [(l - a) / d] + 1

So, n = [(4 - 32) / -4] + 1 = 8

The sum (S) of an arithmetic progression can be found using the formula: S = n/2 * (a + l)

So, S = 8/2 * (32 + 4) = 144

b. 14 : 252

Here, a = 14, l = 252, and d = 252 - 14 = 238

So, n = [(252 - 14) / 238] + 1 = 2

And, S = 2/2 * (14 + 252) = 133

c. 8 : 144

Here, a = 8, l = 144, and d = 144 - 8 = 136

So, n = [(144 - 8) / 136] + 1 = 2

And, S = 2/2 * (8 + 144) = 76

d. 15 : 270

Here, a = 15, l = 270, and d = 270 - 15 = 255

So, n = [(270 - 15) / 255] + 1 = 2

And, S = 2/2 * (15 + 270) = 142.5

e. 7 : 126

Here, a = 7, l = 126, and d = 126 - 7 = 119

So, n = [(126 - 7) / 119] + 1 = 2

And, S = 2/2 * (7 + 126) = 66.5

Please note that the calculations for b, c, d, and e assume that the common difference is the difference between the two given terms, which may not be the case if there are other terms in between.