### Find negative common difference and greatest value of n of the sum being positive of a arithmetic sequence

Question Sample Titled 'Find negative common difference and greatest value of n of the sum being positive of a arithmetic sequence '

The ${1}$st term and the ${11}$th term of an arithmetic sequence are ${605}$ and ${585}$ respectively. Find

 (a) the common difference of the sequence ,  (2 marks) (b) the greatest value of ${n}$ such that the sum of the first ${n}$ terms of the sequence is positive. (3 marks)

 (a) Let ${d}$ be the common difference of the sequence. ${605}+{\left({11}-{1}\right)}{d}$ $={585}$ 1M ${d}$ $=-{2}$ Thus, the common difference of the sequence is $-{2}$ 1A  (b) $\dfrac{{n}}{{2}}{\left[{2}{\left({605}\right)}+{\left({n}-{1}\right)}{\left(-{2}\right)}\right]}$ $>{0}$ 1M $\dfrac{{n}}{{2}}{\left({1210}-{2}{n}+{2}\right)}$ $>{0}$ ${1212}-{2}{n}$ $>{0}$ ${n}>{0}$ ${2}{n}$ $<{1212}$ ${n}$ $<{606}$ 1M ∴   The required greatest value of ${n}$ is ${605}$ . 1A

 Students can also solve the inequality without taking into consideration of ${n}$ being positive at first. Then the inequality would be in quadratic form. (b) $\dfrac{{n}}{{2}}{\left[{2}{\left({605}\right)}+{\left({n}-{1}\right)}{\left(-{2}\right)}\right]}$ $>{0}$ 1M $-{2}{n}^{{2}}+{1212}{n}$ $>{0}$ $-{n}{\left({n}-{606}\right)}$ $>{0}$ ${n}{\left({n}-{606}\right)}$ $<{0}$ ${0}$ $<{n}<{606}$ 1M ∴   The required greatest value of ${n}$ is ${605}$ . 1A

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