The sum 20

The sum of the first six terms of an arithmetic progression is 552, and the sum of the first two terms of the same is 200. Determine the sum of the first 15 terms.

Final Answer:

s₁₅ =  1110

Step-by-step explanation:

$$\begin{gathered} s_6=552 \\ {s}_2=200 \\ S(n)=\frac{n}{2}\cdot (2a+(n-1)d) \\ {s}_6=3\cdot (2\cdot a+5\cdot d) \\ {s}_2=1\cdot (2\cdot a+1\cdot d) \\ 552=3\cdot (2\cdot a+5\cdot d) \\ 200=1\cdot (2\cdot a+1\cdot d) \\ 6a+15d=552 \\ 2a+d=200 \\ Row2-\frac{2}{6}\cdot Row1\to Row2 \\ -4d=16 \\ d=\frac{16}{-4}=-4 \\ a=\frac{552-15d}{6}=\frac{552-15\cdot (-4)}{6}=102 \\ a=102 \\ d=-4 \\ {s}_{15}=\frac{15}{2}\cdot (2\cdot a+(15-1)\cdot d)=\frac{15}{2}\cdot (2\cdot 102+(15-1)\cdot (-4))=1110 \\ \text{ Verifying Solution: } \\ {S}_2=\frac{2}{2}\cdot (2\cdot a+(2-1)\cdot d)=\frac{2}{2}\cdot (2\cdot 102+(2-1)\cdot (-4))=200 \\ {S}_6=\frac{6}{2}\cdot (2\cdot a+(6-1)\cdot d)=\frac{6}{2}\cdot (2\cdot 102+(6-1)\cdot (-4))=552 \end{gathered}$$

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