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Question 33102: a) Determine the sum of the first 20 terms of the following sequence:
4; 8; 16; ...
b) The sum of the first four terms of an arithmetic progression that consists of 20 terms is 21. The sum of the last three terms is 90. Determine the first term and the common difference.
Answer by mukhopadhyay(490)   (Show Source):
You can put this solution on YOUR website! a. Given sequence is a geometric sequence;
Sum of a finite geometric sequence
S = a1(1-r^n)/(1-r) where a1=first term, r=common ration, n=number of terms;
Here, a1 = 4, r=2, n = 20
Sum = 4(1-2^20)/(1-2) = 4(2^10-1)/1 = 4(1023) = 4092;
...........
b. S4 = 21
=> 4(a1+a4)/2 = 21 => a1+a4 = 21/2
=> a1+a4 = 21/2;...........(1)
Sum of last three terms = 90
=> 3(a18+a20)/2 = 90
=> a18+a20 = 60 ........ (2)
From (1): a1+[a1+3d] = 21/2 => 2a1+3d = 21/2 ... (3)
From (2): [a1+17d] + [a1+19d] = 90 = > 2a1+36d = 90 =>a1=45-18d;... (4)
Substituting a1 from (4) into (3):
2(45-18d)+3d=21/2 => 90-33d = 21/2 => 33d = 159/2 => d=53/22
From(4): a1=45-18d => a1= 45-477/11 => a1 = 18/11
Answer: First term is 18/11 and the common difference is 53/22;
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