What is the next term in the sequence: #sqrtx/3, (2sqrtx)/3, sqrtx,...#?

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Answer 1 · 2016-10-24T06:20:41

#(4sqrt(x))/3#

Given:

#sqrt(x)/3, (2sqrt(x))/3, sqrt(x)#

We could also write this as:

#(1sqrt(x))/3, (2sqrt(x))/3, (3sqrt(x))/3#

This is an arithmetic sequence, with common difference #sqrt(x)/3#

So (if it continues as an arithmetic sequence) the next term is formed by adding the common difference.

#(3sqrt(x))/3 + sqrt(x)/3 = (4sqrt(x))/3#

Answer 2 · 2018-01-18T13:57:48

Next term, I.e. the #4^(th)# term #a_4 = color(blue)((4sqrtx)/3)#

Difference between #2^(nd) # & #1^(st)# term is

#(2sqrtx)/3 - sqrtx/3 = sqrtx/3#

Similarly, difference between #3^(rd)# & #2^(nd)# term is #(sqrtx/3)#

Therefore, the common difference between successive terms #d = sqrtx/3#

First term #a = sqrtx/3#

This is an arithmetic progression (A.P) with #a = sqrtx/3, d = sqrtx/3#

#n^th# term of A.P is given by the formula

Sa_n = a + ((n-1)*d) #

Fourth term #a_4 = (sqrtx/3) + ((4-1)*sqrtx/3)#

#a_4 = (sqrtx/3 + (3sqrtx)/3) = (4sqrtx)/3#