what is the common ratio of the geometric progression (√ 2-1)+(3-2 √ 2)+...?
Find the third term of the progression
To find the common ratio of the given geometric progression, we need to divide each term by its preceding term:
(3 - 2√2) ÷ (√2 - 1) = (3 - 2√2) × (√2 + 1) ÷ (√2 - 1) × (√2 + 1)
= (3√2 + 3 - 4√2 - 2) ÷ (2 - 1)
= √2 + 1
So, the common ratio of the given geometric progression is √2 + 1.
To find the third term, we can either calculate it directly using the formula for the nth term of a geometric progression or by multiplying the first two terms by the common ratio:
(√2 - 1) × (√2 + 1) × (√2 + 1)
= (√2 - 1) × (2 + 2√2 + 1)
= (√2 - 1) × (3 + 2√2)
= 3√2 - 5.
Therefore, the third term of the given geometric progression is 3√2 - 5.
a number is made up of two digits. the sum of the digits is 11. if the digits are interchanged, the original number is increased by 9. find the number
Let the original number be 𝑎𝑏, where 𝑎 is the tens digit and 𝑏 is the units digit. From the problem statement, we know that:
𝑎 + 𝑏 = 11 (1)
When the digits are interchanged, the new number is 𝑏𝑎, which is 9 more than the original number:
𝑏𝑎 = 𝑎𝑏 + 9 (2)
We can simplify equation (2) by expanding the products:
10𝑏 + 𝑎 = 10𝑎 + 𝑏 + 9
Simplifying and rearranging terms, we get:
9𝑎 − 9𝑏 = 9
Dividing both sides by 9, we obtain:
𝑎 − 𝑏 = 1 (3)
Now we have two equations ((1) and (3)) with two unknowns (𝑎 and 𝑏) that we can solve simultaneously. Adding equations (1) and (3), we get:
2𝑎 = 12
Therefore, 𝑎 = 6. Substituting this value of 𝑎 into equation (1), we get:
6 + 𝑏 = 11
So 𝑏 = 5. Therefore, the original number is 65.
eight wooden poles are to be used for pillars and the lengths of the poles form an arithmetic progression. if the second pole is 2m and the sixth is 5m, give the lengths of the poles in order
Let the common difference between the consecutive lengths of the poles be 𝑑, and let the length of the second pole (the first term) be 𝑎. Then the lengths of the poles can be expressed as:
𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, 𝑎 + 4𝑑, 𝑎 + 5𝑑, 𝑎 + 6𝑑, 𝑎 + 7𝑑
We know that the length of the second pole is 2m, so 𝑎 + 𝑑 = 2.
Also, the length of the sixth pole is 5m, so 𝑎 + 5𝑑 = 5.
We can use these two equations to solve for 𝑎 and 𝑑. Rearranging the first equation, we get 𝑎 = 2 − 𝑑.
Substituting this into the second equation, we get:
(2 − 𝑑) + 5𝑑 = 5
Expanding and simplifying, we get:
4𝑑 = 3
Therefore, 𝑑 = 0.75.
Substituting this value of 𝑑 into 𝑎 + 𝑑 = 2, we get:
𝑎 + 0.75 = 2
So 𝑎 = 1.25.
Therefore, the lengths of the poles in order are :
1.25m, 2.00m, 2.75m, 3.50m, 4.25m, 5.00m, 5.75m, 6.50m.