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# arithmetic progression

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The fifth term of an arithmetic progression is three times the second term,and the third term is 10.find the 20th term?

Feb 16, 2021

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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?

Hello Guest!

$$\displaystyle a_{1}=a_{1},\ a_{2}=a_{1}+d,\ a_{3}=a_{1}+2d,\ a_{4}=a_{1}+3d, a_5=a_1+4d\dots$$

$${\displaystyle a_{i}=a_{1}+(i-1)\cdot d\quad } \ (explicit\ formula,\ explizite\ Formel)$$

$$a_5=(a_1+d)\cdot 3=a_1+(5-1)\cdot d\ (specification\ 1 )\\ a_3=a_1+(3-1)\cdot d=10\ (specification\ 2 )$$

$$3a_1+3d=a_1+4d\\ d=2a_1$$

$$a_1+2d=10\\ d=\dfrac{10-a_1}{2}$$

$$2a_1=\dfrac{10-a_1}{2}\\ 4a_1=10-a_1$$

$$\large a_1=2$$

$$d=\dfrac{10-a_1}{2}\\ d=\dfrac{10-2}{2}\\$$

$$\large d=4$$

$$a_{i}=a_{1}+(i-1)\cdot d\\ a_{20}=2+(20-1)\cdot 4\\$$

$$\large a_{20}=78$$

!

Feb 16, 2021
edited by asinus  Feb 16, 2021