Question

Find the sum of the following arithmetic progressions:

(i) 50, 46, 42, ... to 10 terms

(ii) 1, 3, 5, 7, ... to 12 terms

(iii) 3, 9/2, 6, 15/2, ... to 25 terms

(iv) 41, 36, 31, ... to 12 terms

(v) a + b, a − b, a − 3b, ... to 22 terms

(vi) (x − y)², (x² + y²), (x + y)², ..., to n terms

(vii) $\frac{x-y}{x+y}\frac{3x-2y}{x+y}\frac{5x-3y}{x+y},...$to n terms

(viii) −26, −24, −22, ... to 36 terms.

Answer 1

In the given problem, we need to find the sum of terms for different arithmetic progressions. So, here we use the following formula for the sum of n terms of an A.P.,

Where; a = first term for the given A.P.

d = common difference of the given A.P.

n = number of terms

(i) To 10 terms

Common difference of the A.P. (d)

=

Number of terms (n) = 10

First term for the given A.P. (a) = 50

So, using the formula we get,

Therefore, the sum of first 10 terms for the given A.P. is.

(ii) To 12 terms.

Common difference of the A.P. (d)

=

Number of terms (n) = 12

First term for the given A.P. (a) = 1

So, using the formula we get,

Therefore, the sum of first 12 terms for the given A.P. is.

(iii) To 25 terms.

Common difference of the A.P. (d) =

Number of terms (n) = 25

First term for the given A.P. (a) = 3

So, using the formula we get,

On further simplifying, we get,

Therefore, the sum of first 25 terms for the given A.P. is.

(iv) To 12 terms.

Common difference of the A.P. (d) =

Number of terms (n) = 12

First term for the given A.P. (a) = 41

So, using the formula we get,

Therefore, the sum of first 12 terms for the given A.P. is.

(v) To 22 terms.

Common difference of the A.P. (d) =

Number of terms (n) = 22

First term for the given A.P. (a) =

So, using the formula we get,

Therefore, the sum of first 22 terms for the given A.P. is.

(vi) To n terms.

Common difference of the A.P. (d) =

Number of terms (n) = n

First term for the given A.P. (a) =

So, using the formula we get,

Now, taking 2 common from both the terms inside the bracket we get,

Therefore, the sum of first n terms for the given A.P. is

(vii) To n terms.

Number of terms (n) = n

First term for the given A.P. (a) =

Common difference of the A.P. (d) =

So, using the formula we get,

Now, on further solving the above equation we get,

Therefore, the sum of first n terms for the given A.P. is.

(viii) To 36 terms.

Common difference of the A.P. (d) =

Number of terms (n) = 36

First term for the given A.P. (a) = −26

So, using the formula we get,

Therefore, the sum of first 36 terms for the given A.P. is.