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

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Answer 1 · 2018-06-01T00:42:20

The first term is #35/18#

The common difference is #32/9 #

Suppose we denote the #n^(th)# term of the AP by #u_n# where #u_n=a+nd#, making the AP sequence:

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

Given the sum of the first and last term is #43#, we have:

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

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

Given, also, that the third term is #11#, we have:

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

Given, also, that the sum is #258#, and utilising the AP summation formula:

# 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 #n=12# into [A] we get:

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

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

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

# :. d = 32/9 #

Finally, substituting #d=32/9# into [D]:

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

# :. a = 35/18#