Common difference

The 4th term of an arithmetic progression is 6. If the sum of the 8th and 9th term is -72, find the common difference.

Correct result:

d =  -9.3333

Solution:

$$\begin{gathered} a_4=6 \\ {a}_8+a_9=-72 \\ (a_4+4d)+(a_4+5d)=-72 \\ 2a_4+9d=-72 \\ d=(-72-2\cdot a_4)\mathrm{/}9=(-72-2\cdot 6)\mathrm{/}9=-{\displaystyle \frac{28}{3}}=-9\frac{1}{3}\doteq -9.3333=-9.3333 \\ \text{ Verifying Solution: } \\ {a}_1=a_4-3\cdot d=6-3\cdot (-9.3333)=34 \\ {k}_8=a_1+7\cdot d=34+7\cdot (-9.3333)=-{\displaystyle \frac{94}{3}}=-31\frac{1}{3}\doteq -31.3333 \\ {k}_9=a_1+8\cdot d=34+8\cdot (-9.3333)=-{\displaystyle \frac{122}{3}}=-40\frac{2}{3}\doteq -40.6667 \\ {s}_{89}=k_8+k_9=(-31.3333)+(-40.6667)=-72 \end{gathered}$$

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