document.write( "Question 936199: The ninth term of an arithmetic progression is 52 and the sum of the first twelve terms is 414.Find the first term and the common difference. \n" ); document.write( "

Algebra.Com's Answer #569636 by srinivas.g(540) $\"\"$ $\"About$

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formula:
\n" ); document.write( " nth term =a+(n-1)d
\n" ); document.write( " sum of n terms = $\"%28n%2F2%29%282a%2B%28n-1%29d%29\"$
\n" ); document.write( " where d = common difference , a = first term
\n" ); document.write( " 9 th term = a+(9-1)d
\n" ); document.write( " 52 = a+8d...................eq(1)
\n" ); document.write( " sum of fisrt 12 terms = 414
\n" ); document.write( " $\"%2812%2F2%29%282a%2B%2812-1%29d%29+=414\"$
\n" ); document.write( " $\"6%282a%2B11d%29=414\"$
\n" ); document.write( " divide with 6 on both sides
\n" ); document.write( " $\"+6%282a%2B11d%29%2F6+=414%2F6\"$
\n" ); document.write( " $\"2a%2B11d+=+69\"$ .............eq(2)
\n" ); document.write( " we need to solve eq(1) & eq(2)
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Solved by pluggable solver: SOLVE linear system by SUBSTITUTION

Solve:
\n" ); document.write( " $\"+system%28+%0D%0A++++1%5Ca+%2B+8%5Cd+=+52%2C%0D%0A++++2%5Ca+%2B+11%5Cd+=+69+%29%0D%0A++\"$ We'll use substitution. After moving 8*d to the right, we get:
\n" ); document.write( " $\"1%2Aa+=+52+-+8%2Ad\"$ , or $\"a+=+52%2F1+-+8%2Ad%2F1\"$ . Substitute that
\n" ); document.write( " into another equation:
\n" ); document.write( " $\"2%2A%2852%2F1+-+8%2Ad%2F1%29+%2B+11%5Cd+=+69\"$ and simplify: So, we know that d=7. Since $\"a+=+52%2F1+-+8%2Ad%2F1\"$ , a=-4.
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\n" ); document.write( " Answer: $\"system%28+a=-4%2C+d=7+%29\"$ .
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