document.write( "Question 1121963: An arithmetic sequence a1,a2,......a100 has a sum of 15 000. Finf the first term and the common difference if the sum of the terms in the sequence a3, a6, a9,... a99 is 5016. \n" ); document.write( "

Algebra.Com's Answer #737987 by greenestamps(13203) $\"\"$ $\"About$

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\n" ); document.write( "The sum of the first 100 terms is 15000, so the average of the terms is 15000/100 = 150.

\n" ); document.write( "In particular, the average of the first and last terms is 150; and the sum of the 50th and 51st terms is 150.

\n" ); document.write( "The sub-sequence has 33 terms with a sum of 5016, so the average is 5016/33 = 152.

\n" ); document.write( "In particular, the 17th (middle) term alone is 152.

\n" ); document.write( "The 17th term of the sub-sequence is the 51st term of the full sequence.

\n" ); document.write( "So we know the average of the 50th and 51st terms is 150; and we know that the 51st term is 152. That means the 50th term is 148.

\n" ); document.write( "With the 50th term 148 and the 51st term 152, the common difference is 4.

\n" ); document.write( "With a common difference of 4 and 51st term 152, the first term is 152-50(4) = 152-200 = -48.

\n" ); document.write( "Answer: first term -48; common difference 4.

\n" ); document.write( "CHECK:
\n" ); document.write( "a1+a2+...+a100 = (-48)+(-44)+...+(348) = 100((-48)+348)/2 = 100(150) = 15000
\n" ); document.write( "a3+a6+...+a99 = (-40)+(-28)+...+344) = 33((-40)+344)/2 = 33(152) = 5016 \n" ); document.write( "

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