# What is the common difference? if an A. P has the sum of first and last term is 43 and the third is 11 the sum is 258

##### 1 Answer

The first term is

The common difference is

#### Explanation:

Suppose we denote the

# {a, a+d, a+2d, a+3d, ..., a+(n-1)d }#

Given the sum of the first and last term is

# (a) + (a+(n-1)d) = 43 #

# :. 2a+(n-1)d = 43 # ..... [A]

Given, also, that the third term is

# a + 2d = 11 # ..... [B]

Given, also, that the sum is

# S_n = n/2(2a+(n-1)d)#

We have:

# n/2(2a+(n-1)d) = 258 #

# n(2a+(n-1)d) = 516 # ..... [C]

Substituting [A] into [C] we get:

# n(43) = 516 => n =12 #

Back substitution of

# :. 2a+11d = 43 # ..... [D]

So that [D]-2[B] yields:

# 11d-2d = 43-11 => 9d = 32#

# :. d = 32/9 #

Finally, substituting

# 2a+(11*32)/9 = 43 => 2a = 43 - 352/9#

# :. a = 35/18#