The fifth term of an arithmetic progression is three times the second term,and the third term is 10.find the 20th term?
+15001 The fifth term of an arithmetic progression is three times the second term,and the third term is 10.find the 20th term?
Der fünfte Term einer arithmetischen Folge ist dreimal so groß wie der zweite Term und der dritte Term ist 10. Finden Sie den 20. Term?
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\(\displaystyle a_{1}=a_{1},\ a_{2}=a_{1}+d,\ a_{3}=a_{1}+2d,\ a_{4}=a_{1}+3d, a_5=a_1+4d\dots \)
\({\displaystyle a_{i}=a_{1}+(i-1)\cdot d\quad } \ (explicit\ formula,\ explizite\ Formel)\)
\(a_5=(a_1+d)\cdot 3=a_1+(5-1)\cdot d\ (specification\ 1 )\\ a_3=a_1+(3-1)\cdot d=10\ (specification\ 2 )\)
\(3a_1+3d=a_1+4d\\ d=2a_1\)
\(a_1+2d=10\\ d=\dfrac{10-a_1}{2}\)
\(2a_1=\dfrac{10-a_1}{2}\\ 4a_1=10-a_1\)
\(\large a_1=2\)
\( d=\dfrac{10-a_1}{2}\\ d=\dfrac{10-2}{2}\\ \)
\(\large d=4\)
\(a_{i}=a_{1}+(i-1)\cdot d\\ a_{20}=2+(20-1)\cdot 4\\\)
\(\large a_{20}=78\)
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