Question

Hola !

question for 100 points


(1 ) find the sum 7 + 77 + 777 + 7777 ...... to nth terms .

(2) if a , b, c are in Gp and a + x , b+ x , c + x are in H.P then prooved that x = b

Answer 1

Hello dear !!!

1st solution :-

7 + 77 + 777 + 7777 ...... to nth terms

=> 7 ( 1 + 11 + 111 + 1111 + 1111.... to nth terms )

multiplying and dividing by 9

=> 7/9( 9 + 99 + 999 +...... to nth term )

=> 7/9 [( 10 -1 ) ( 10² - 1) + ( 10³ + 1) + ( 10⁴ - 1) ....( 10^n - 1) ]

we know that sum of gp formula

sn = a (r^n - 1) / r - 1

using similarly

=> 7/9 [ {10 (10^n - 1 ) / 10 - 1 } - n]

=> 7 [ 10^n+1 - 10 - 9n/ 81] Answer

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2nd solution .

if a , b, c are in are in Gp

then , b² = ac

and , if a+x , b+ x , c + x, are in H.P

then, b + x = 2 ( a + x ) ( c + x) /(a + x )+(c + x)

( b+ x) ( a + c + 2x ) = 2 ( a + x ) ( c + x)

ab + ac + bc + cx + 2bx + 2x² = 2ac + 2ax + 2cx + 2x²

x ( 2b - a - c) = 2ac - ab - bc

x ( 2b - a - c) = b( 2b - a - c )

x = b prooved ♻

________________________

Hope it helps you !!!

@Rajukumar111

Answer 2

______✨ HEY MATE ✨______

➡️Here is your SOLUTION ⤵️

7 + 77 + 777 + 7777 ...... to nth terms

=> 7 ( 1 + 11 + 111 + 1111 + 1111.... to nth terms )

multiplying and dividing by 9

=> 7/9( 9 + 99 + 999 +...... to nth term )

=> 7/9 [( 10 -1 ) ( 10² - 1) + ( 10³ + 1) + ( 10⁴ - 1) ....( 10^n - 1) ]

we know that sum of gp formula

sn = a (r^n - 1) / r - 1

using similarly

=> 7/9 [ {10 (10^n - 1 ) / 10 - 1 } - n]

=> 7 [ 10^n+1 - 10 - 9n/ 81] Answer

✌️ I THINK IT HELPED YOU ✌️

➡️ @dmohit432 ✔️