Question

Дано (an)-арефметичної прогресії а1=7 а2=15 знайти а8 , s12-?

Answer 1

Ответ:

Арифметическая прогрессия :   $\bf a_1=7\ ,\ \ a_2=15$  .

Формулы :  $\bf a_{n}=a_1+d(n-1)\ \ ,\ \ d=a_{n+1}-a_{n}\ \ ,\ \ S_{n}=\dfrac{a_1+a_{n}}{2}\cdot n$   .  

$\bf d=a_2-a_1=16-7=8\\\\a_8=a_1+7d=7+7\cdot 8=\boxed{\bf 63}$  

$\bf a_{12}=a_1+11\, d=7+11\cdot 8=1+88=95\\\\S_{12}=\dfrac{7+95}{2}\cdot 11=\dfrac{102}{2}\cdot 11=51\cdot 11=\boxed{\bf 561}$

Answer 2

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. In this case, we are given an arithmetic progression with the first term (a1) equal to 7 and the second term (a2) equal to 15. We need to find the eighth term (a8) and the sum of the first twelve terms (s12).

Finding the Eighth Term (a8)

To find the eighth term (a8) of an arithmetic progression, we can use the formula:

an = a1 + (n - 1)d

where: - an is the nth term of the arithmetic progression - a1 is the first term of the arithmetic progression - n is the position of the term we want to find - d is the common difference between consecutive terms

In this case, we are looking for the eighth term (a8), so n = 8. The first term (a1) is given as 7. To find the common difference (d), we can subtract the first term (a1) from the second term (a2):

d = a2 - a1

Let's calculate the common difference (d) and then use the formula to find the eighth term (a8):

d = 15 - 7 = 8

a8 = a1 + (8 - 1)d = 7 + 7(8) = 7 + 56 = 63

Therefore, the eighth term (a8) of the arithmetic progression is 63.

Finding the Sum of the First Twelve Terms (s12)

To find the sum of the first twelve terms (s12) of an arithmetic progression, we can use the formula:

s12 = (n/2)(a1 + an)

where: - s12 is the sum of the first twelve terms - n is the number of terms we want to sum (in this case, n = 12) - a1 is the first term of the arithmetic progression - an is the nth term of the arithmetic progression

Let's use the formula to find the sum of the first twelve terms (s12):

s12 = (12/2)(a1 + a12)

To find the twelfth term (a12), we can use the formula:

a12 = a1 + (12 - 1)d

where d is the common difference we calculated earlier.

a12 = 7 + (12 - 1)8 = 7 + 11(8) = 7 + 88 = 95

Now we can substitute the values into the formula to find the sum of the first twelve terms (s12):

s12 = (12/2)(a1 + a12) = 6(7 + 95) = 6(102) = 612

Therefore, the sum of the first twelve terms (s12) of the arithmetic progression is 612.

I hope this helps! Let me know if you have any further questions.