Question

9+17+25+...+(8n+1)=125 знайти n

Answer 1

Task/28609154
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9+17+25+ ...+(8n+1) =125
(8*1+1)+(8*2+1) +(8*3+1)+ ...+(8n+1) = 125 ;
8(1+2+3+...+n) +(1+1+1+...+1) =125 ;
8(1+n)n/2 + n =125 ;
4n² +5n -125 =0 ;   D =5² -4*4*(-125) =25 +2000=2025 =45²
n₁ = (-5-45)/8= -25/4 _посторонний корень
n₂ =(-5+45)/8 = 5.

ответ: 5.
* * * * * * * * * * * * * * * * * * * * * * * * * * 
Или как сумма арифметической прогрессии с a₁ =9 , d =8;
(2*9+(n-1)*8 )n/2 =125 ;
2(9+(n-1)*4 )n/2 =125 ;
(9 +4n-4)n =125 ;
4n²+5n -125=0 

Answer 2

To find the value of n in the equation:

9 + 17 + 25 + ... + (8n + 1) = 125

We need to find the value of n that satisfies the equation. Let's find the sum of the arithmetic sequence and set it equal to 125:

The sum of an arithmetic sequence is given by the formula: Sum = (n/2) * (first term + last term)

Here, the first term (a) is 9 and the last term (8n + 1) is the nth term. So, the sum is:

Sum = (n/2) * (9 + (8n + 1))

Now, we can set this sum equal to 125:

(n/2) * (9 + 8n + 1) = 125

Now, simplify the equation:

(n/2) * (8n + 10) = 125 4n^2 + 5n - 250 = 0

Now, we have a quadratic equation. We can solve it using the quadratic formula:

n = (-b ± √(b^2 - 4ac)) / 2a

where a = 4, b = 5, and c = -250.

n = (-(5) ± √(5^2 - 4 * 4 * -250)) / (2 * 4) n = (-5 ± √(25 + 4000)) / 8 n = (-5 ± √4025) / 8 n = (-5 ± 63.436)/8

Now, we have two possible solutions for n:

1. n = (-5 + 63.436) / 8 n ≈ 58.436 / 8 n ≈ 7.3055

2. n = (-5 - 63.436) / 8 n ≈ -68.436 / 8 n ≈ -8.5545

Since n should be a positive integer (as it represents the number of terms in the sequence), we can discard the negative solution. Thus, the value of n that satisfies the equation is approximately n ≈ 7.3055. However, since the number of terms in the sequence must be a whole number, we should round down to the nearest integer:

n = 7

So, when n = 7, the sum of the sequence will be equal to 125.