How do I find the sum of the terms of this sequence?

Question

The ratio of the fifth term to the twelfth term of a sequence in an arithmetic progression is 6/13. If each term of this sequence is positive, and the product of the first term and the third term is 32, what is the sum of the first 100 terms of this sequence?

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Answer 1 · 2018-08-04T01:07:15

1/41 { 576 656 736 816 896 976 1056 1136 1216 1296
1376 1456 ... }
Sum of the st 100 terms = 453600/41 = 11063 + 17/41

Let the sequence be S #{t_k), k = 1, 2, 3, ...#.

#t_k = a + ( k - 1 ) d#

#t_5/t_12 = ( a + 4 d ) / ( a + 11 d ) = 8/13#. So,

#5 a = 36 d#

#t_1 + t_3 = 2 a + 2d = 32#> So,

#a + d = 16. solving,

#a =( (16)(36) )/41= 576/41

#d = (5/36)( (16)(36) )/41= 80/41#. And so,

S = 1/41 { 576 656 736 816 896 976 1056 1136 1216 1296 #

1376 1456 ... }

Sum of 1st 100 terms #= 100 a + 1/2( 99 ) ( 100 ) d#

#= 100(576/41) + 4950 ( 80/41 ) = 453600/41#