Question

Помогите пж срочно. 9 класс 1) В арифметической прогрессии а7+а1=4, а3^2+а7^2=122. Найти а1 и d. 2)В арифметической прогрессии сумма членов со 2 по 5 включительно равна 34, а разность 2 и 3 равна 28. Найти а1 и d.

Answer 1

1
{a7+a1=4⇒a1+6d+a1=4⇒2a1+6d=4⇒a1+3d=2⇒a1=2-3d
{a3²+a7²=122⇒(a1+2d)²+(a1+6d)²=122⇒(2-d)²+(2+3d)²=122
4-4d+d²+4+12d+9d²-122=0
10d²+8d-114=0
5d²+4d-57=0
D=16+1140=1156
d1=(-4-34)/10=-3,8⇒a1=2-3*(-3,8)=2+11,4=13,4
d2=(-4+34)/10=3⇒a1=2-3*3=2-9=-7
2
{a2+a3+a4+a5=34⇒4a1+10d=34⇒2a1+5d=17
{a2-a3=28⇒a1+d-a1-2d=28⇒-d=28⇒d=-28
2a1-140=17
2a1=157
a1=78,5

Answer 2

Problem 1: Finding a1 and d in an arithmetic progression

To find the values of a1 and d in an arithmetic progression, we can use the given information.

Given: - a7 - a1 = 4 - a3^2 + a7^2 = 122

Let's solve this step by step.

Step 1: Finding the common difference (d) We know that the difference between any two consecutive terms in an arithmetic progression is constant. Let's assume the common difference is d.

Since a7 - a1 = 4, we can write it as: a1 + 6d - a1 = 4 6d = 4 d = 4/6 d = 2/3

So, the common difference (d) is 2/3.

Step 2: Finding the first term (a1) To find a1, we need to substitute the value of d into the equation a7 - a1 = 4.

a7 - a1 = 4 (a1 + 6d) - a1 = 4 6d = 4 6(2/3) = 4 4 = 4

Since the equation is satisfied, we can conclude that the given information is consistent.

Therefore, the first term (a1) is not uniquely determined by the given information. It can take any value.

The common difference (d) is 2/3.

Problem 2: Finding a1 and d in an arithmetic progression

To find the values of a1 and d in an arithmetic progression, we can use the given information.

Given: - The sum of terms from the 2nd term to the 5th term (inclusive) is 34. - The difference between the 2nd term and the 3rd term is 28.

Let's solve this step by step.

Step 1: Finding the common difference (d) We know that the difference between any two consecutive terms in an arithmetic progression is constant. Let's assume the common difference is d.

Since the difference between the 2nd term and the 3rd term is 28, we can write it as: a2 - a3 = 28 (a1 + d) - (a1 + 2d) = 28 -d = 28 d = -28

So, the common difference (d) is -28.

Step 2: Finding the first term (a1) To find a1, we need to use the sum of terms from the 2nd term to the 5th term.

The sum of terms from the 2nd term to the 5th term (inclusive) is 34. We can write it as: (a1 + d) + (a1 + 2d) + (a1 + 3d) + (a1 + 4d) = 34

Simplifying the equation: 4a1 + 10d = 34 4a1 + 10(-28) = 34 4a1 - 280 = 34 4a1 = 314 a1 = 314/4 a1 = 78.5

So, the first term (a1) is 78.5.

The common difference (d) is -28.

Please note that in an arithmetic progression, the terms are usually integers. However, in this case, the first term (a1) is a decimal number.