5 uždavinys
Didėjančios geometrinės progresijos pirmasis narys lygus 2, o trečiasis lygus 18. Antrasis šios progresijos narys lygus:
A- 6 B 6 C 9 D 10
Sprendimas.
$$b_{3} = b_{2}\cdot q$$ (1)
$$b_{2} = b_{1}\cdot q$$ (2)
Pirmą lygtį padaliname iš antrosios:
| b_3 |
| / b_2 |
| b_2* q |
| / b_1/ q |
|
| b_3 |
| / b_2 |
| b_2* q |
| / b_1/ q |
| b_3 |
| / b_2 |
| b_2 |
| / b_1 |
b_3 =
$$b_{3}$$ = $$\frac{b_{2}\cdot b_{2}}{b_{1}}$$
| b_2* b_2 |
| / b_1 |
b_1* b_3 = b_2* b_2$$b_{1}\cdot b_{3}$$ = $$b_{2}\cdot b_{2}$$
b_1* b_3 = b_2^2$$b_{1}\cdot b_{3}$$ = $$b_{2}^{2}$$
b_2^2 = b_1* b_3$$b_{2}^{2}$$ = $$b_{1}\cdot b_{3}$$
saknis( b_2^2) = saknis( b_1* b_3)$$\sqrt {b_{2}^{2}}$$ = $$\sqrt {b_{1}\cdot b_{3}}$$
b_2^2) = saknis( b_1* b_3)$$\sqrt {b_{2}^{2}}$$ = $$\sqrt {b_{1}\cdot b_{3}}$$b_2 = saknis( b_1* b_3)$$b_{2}$$ = $$\sqrt {b_{1}\cdot b_{3}}$$
b_1* b_3)$$b_{2}$$ = $$\sqrt {b_{1}\cdot b_{3}}$$ Paaiškinimas: b_2 = saknis( 2* b_3)$$b_{2}$$ = $$\sqrt {2\cdot b_{3}}$$
2* b_3)$$b_{2}$$ = $$\sqrt {2\cdot b_{3}}$$ Paaiškinimas: b_2 = saknis( 2* 18)$$b_{2}$$ = $$\sqrt {2\cdot 18}$$
2* 18)$$b_{2}$$ = $$\sqrt {2\cdot 18}$$b_2 = saknis(36)$$b_{2}$$ = $$\sqrt {36}$$
36)$$b_{2}$$ = $$\sqrt {36}$$b_2 = 6$$b_{2}$$ = $$6$$
$$\frac{b_{3}}{b_{2}}$$ = $$\frac{b_{2}\cdot q}{b_{1}\cdot q}$$
$$\frac{b_{3}}{b_{2}}$$ = $$\frac{b_{2}}{b_{1}}$$
$$b_{2}^{2}$$ = $$b_{1}\cdot b_{3}$$
$$b_{2}$$ = $$\sqrt {b_{1}\cdot b_{3}}$$
$$b_{2}$$ = $$\sqrt {2\cdot 18}$$
$$b_{2}$$ = $$\sqrt {36}$$
$$b_{2}$$ = $$6$$
Atsakymas: B