Question

a) (7^1×7^6×7^11×....×7^301) : (7^1×7^4×7^7×....×7^154)

b) (4^1×4^2×4^3×....×4^150) : (2^1×2^3×2^5×....×2^251)

c) (9^2×9^4×9^6×....×9^80) : (3^1×3^2×3^3×....×3^50)

Plz e urgent! Coroniță!!

Answer 1

a)  ( 7¹ x 7⁶ x 7¹¹ x ..... x 7³⁰¹ ) : ( 7¹ x 7⁴ x 7⁷ x ........x 7¹⁵⁴ ) =

= 7 ⁽ ¹ ⁺ ⁶ ⁺ ¹¹ ⁺ .... ⁺ ³⁰¹ ⁾ :  7 ⁽ ¹ ⁺ ⁴ ⁺ ⁷ ⁺ ...... ⁺ ¹⁵⁴ ⁾

1 + 6 + 11 + .... + 301 =

→ stabilesc cati termeni are suma cu ratia 5

( 301 - 1 ) : 5 + 1 - 300 : 5 + 1 = 61 termeni are suma

→ aplic formula sumei lui Gauss

= 61 × ( 1 + 301 ) : 2 =

= 61 × 302 : 2 =

= 61 × 151 =

= 9 211

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1 + 4 + 7 + ..... + 154 = 52 × ( 1 + 154 ) : 2 = 26 × 155 = 4 030

( 154 - 1 ) : 3 + 1 = 153 : 3 + 1 = 52 termeni are suma

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= 7 ⁹²¹¹ : 7⁴⁰³⁰ = 7 ⁽ ⁹²¹¹ ⁻ ⁴ ⁰³⁰ ⁾ = 7⁵¹⁸¹

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b)

( 4¹ x 4² x 4³x ..... x 4¹⁵⁰  ) : ( 2¹ x 2³ x 2⁵ x .... x 2²⁵¹ ) =

= 4 ⁽ ¹⁺ ²⁺³⁺...... ⁺¹⁵⁰⁾  : 2 ⁽ ¹⁺³⁺⁵⁺... ⁺²⁵¹ ⁾=

= 4¹¹³²⁵ : 2¹⁵⁸⁷⁶ =

= ( 2² ) ¹¹³²⁵ : 2¹⁵⁸⁷⁶ =

= 2 ²²⁶⁵⁰ : 2 ¹⁵⁸⁷⁶ =

= 2 ⁶⁷⁷⁴

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1 + 2 + 3 +.......+  150 = 150 × ( 1 + 150 ) : 2 = 75 × 151 = 11 325

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1 + 3 + 5 + ...+ 251 = 126 × ( 1 + 251 ) : 2 = 63 × 252 = 15 876

( 251 - 1) : 2 + 1 = 250 : 2 = 126 termeni are suma

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c)

( 9² x 9⁴ x 9⁶ x .... x 9⁸⁰ ) : ( 3¹ x 3² x 3³ x ...... x 3⁵⁰ ) =

= 9 ² ˣ ⁽ ¹ ⁺²⁺³⁺..... ⁺⁴⁰⁾  :  3⁽ ¹ ⁺ ²⁺³⁺...... ⁺⁵⁰ ⁾ =

= 9¹⁶⁴⁰ : 3 ¹²⁷⁵ =

= ( 3² ) ¹⁶⁴⁰ : 3¹²⁷⁵ =

= 3 ⁽ ³²⁸⁰ ⁻ ¹²⁷⁵⁾ =

= 3²⁰⁰⁵

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2 x ( 1 + 2 + 3 + ,,,, + 40 ) = 2 x 40 x ( 1 + 40 ) : 2 = 40 x 41 = 1 640

1 + 2 + 3 + .... + 50 = 50 x ( 1 + 50 ) : 2 = 25 x 51 = 1 275