Question

Если S4=-28,s6=58 найдите S16(арифметическая прогрессия)

Answer 1

По формуле суммы n первых членов арифметической прогрессии:
   $S_4= \dfrac{2a_1+3d}{2}\cdot 4=4a_1+6d \\ \\ \\ S_6= \dfrac{2a_1+5d}{2}\cdot 6 =6a_1+15d$

$\displaystyle \left \{ {{2a_1+6d=-28} \atop {6a_1+15d=58}} \right. ~~\Rightarrow~~~ \left \{ {{2a_1+3d=-14} \atop {6a_1+15d=58}} \right. ~~\Rightarrow~~ \left \{ {{a_1=\frac{-14-3d}{2}} \atop {6\cdot\frac{-14-3d}{2}+15d=58}} \right.$


$3\cdot(-14-3d)+15d=58\\ \\ -42-9d+15d=58\\\\ 6d=100\\ \\ d= \frac{50}{3};$         $a_1=-32;$


Тогда сумма первых шестнадцати членов а.п., равна:
   
              $S_{16}= \dfrac{2a_1+15d}{2}\cdot16=8(2a_1+15d)=8\cdot(2\cdot(-32)+15\cdot \frac{50}{3} ) =1488$

Answer 2

Arithmetic Progression Calculation

To find the value of S16 in the arithmetic progression given that S4 = -28 and S6 = 58, we can use the formula for the sum of the first n terms in an arithmetic progression:

Sn = n/2 * (a + l)

Where: - Sn is the sum of the first n terms - a is the first term - l is the last term - n is the number of terms

First, we need to find the common difference (d) in the arithmetic progression. We can use the given terms S4 and S6 to find d.

The formula for the nth term of an arithmetic progression is: An = a + (n-1)d

Using the given values: 1. S4 = -28 implies A4 = -28 2. S6 = 58 implies A6 = 58

We can use these to find the common difference d.

Calculations

Using the formula An = a + (n-1)d: 1. A4 = a + 3d = -28 2. A6 = a + 5d = 58

Subtracting the equations, we get: (A6 - A4) = 2d = 86

So, d = 43.

Now that we have the common difference, we can find the first term a using the equation A4 = a + 3d: -28 = a + 3*43 -28 = a + 129 a = -157

Finally, we can find S16 using the formula Sn = n/2 * (a + l), where n = 16, a = -157, and l can be found using l = a + (n-1)d.

Conclusion

Using the calculated values, we find that S16 = -157 + 15*43 = 578.