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Lizza fobb

asked 2017-01-21 22:16:35

Sum of the series in arithmetic progression is $72.$ The first term is $17$ and the common difference is $-2.$ Find the number of terms.

jiju

answered 2017-01-21 22:41:56

It is easy to see $17+15+13+11+9+7=72$ and therefore number of terms is $6$. See the following solution using the formulas of arithmetic progression.

$S_n=\dfrac{n}{2}[2a+(n-1)d]\\

72=\dfrac{n}{2}[2×17+(n-1)(-2)]\\

72=\dfrac{n}{2}(34-2n+2)\\

72=\dfrac{n}{2}(36-2n)\\

72=n(18-n)\\

72=18n-n^2\\

n^2-18n+72=0\\

(n-6)(n-12)=0\\

n=6\text{ or }12$

So, number of terms can be $6$ or $12$

If number of terms is $6$,

$17+15+13+11+9+7=72$

If number of terms is $12$,

$17+15+13+11+9+7+5$ $+3+1+(-1)+(-3)+(-5)=72$

$S_n=\dfrac{n}{2}[2a+(n-1)d]\\

72=\dfrac{n}{2}[2×17+(n-1)(-2)]\\

72=\dfrac{n}{2}(34-2n+2)\\

72=\dfrac{n}{2}(36-2n)\\

72=n(18-n)\\

72=18n-n^2\\

n^2-18n+72=0\\

(n-6)(n-12)=0\\

n=6\text{ or }12$

So, number of terms can be $6$ or $12$

If number of terms is $6$,

$17+15+13+11+9+7=72$

If number of terms is $12$,

$17+15+13+11+9+7+5$ $+3+1+(-1)+(-3)+(-5)=72$

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