Question

A18=-20 S10=-40 a1-? АРИФМЕТРИЧНА ПРОГРЕСІЯ

Answer 1

$a_{18}=-20\\\\a_{1}+17d=-20\\\\a_{1} =-17d-20\\\\S_{10} =\dfrac{2a_{1} +9d}{2}\cdot 10\\\\S_{10} =(2a_{1}+9d)\cdot 5\\\\-40=(-17d-20+9d)\cdot 5\\\\-8d-20=-8\\\\-8d=12\\\\d=-1,5\\\\a_{1}=-17\cdot (-1,5)-20=25,5-20=5,5\\\\\boxed{a_{1}=5,5}$

Answer 2

To find the first term ($a_1$) of an arithmetic progression when you know the 18th term ($A_{18} = -20$) and the 10th term ($S_{10} = -40$), you can use the formula for the $n$th term of an arithmetic progression:

$A_n = a_1 + (n - 1) \times d$

where $A_n$ is the $n$th term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference between consecutive terms.

First, use the given information to find the common difference ($d$). You know that the 18th term is -20:

$A_{18} = a_1 + 17d = -20$

And the 10th term is -40:

$A_{10} = a_1 + 9d = -40$

Now, subtract the second equation from the first equation to eliminate $a_1$:

$(a_1 + 17d) - (a_1 + 9d) = -20 - (-40)$ $8d = -20 + 40$ $8d = 20$ $d = 2.5$

Now that you know the common difference ($d = 2.5$), you can find the first term ($a_1$) using either of the original equations. Let's use the one with the 10th term:

$a_1 + 9d = -40$ $a_1 + 9(2.5) = -40$ $a_1 + 22.5 = -40$ $a_1 = -40 - 22.5$ $a_1 = -62.5$

So, the first term ($a_1$) of the arithmetic progression is $-62.5$.