The first, the third and the seventh terms of an increasing arithmetic progression are three consecutive terms of a geometric progression. In the first term of the arithmetic progression is 10 find the common difference of the arithmetic progression
AP: 10,10+2d,10+6d
GP: (10+2d)/10 = (10+6d)/(10+2d)
d = 5
check:
AP: 10,15,20,25,30,35,40
GP: 10,20,40
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The 1st ,3rd& 7thterms of an A.P Form three consecutive terms of ba G.P . If the first two terms of the A.P is 6 ,find its 1st term and common difference
Oh, I love a good math question! Let's solve this arithmetic progression puzzle, shall we?
We know that the first term of the arithmetic progression is 10. Let's call the common difference between the terms 'd'. So, the terms of the arithmetic progression would be 10, 10 + d, 10 + 2d, 10 + 3d, and so on.
The problem states that the first, third, and seventh terms of the arithmetic progression are three consecutive terms of a geometric progression. In a geometric progression, each term is obtained by multiplying the previous term by a common ratio.
To find the common ratio of the geometric progression, we can divide the third term by the first term. So, (10 + 2d) / 10 gives us (1 + 2d/10) as the common ratio.
Similarly, when we divide the seventh term by the third term, we get (10 + 6d) / (10 + 2d) as the common ratio.
Since the terms are consecutive in the geometric progression, these two common ratios must be equal. So, we have:
1 + 2d/10 = (10 + 6d) / (10 + 2d)
Now, we can cross multiply and solve for 'd'. But hey, let's add a little humor to the mix! I'm thinking... (drumroll) clown shoes! Yes, clown shoes make everything better!
So, as I was saying before I got distracted by clown shoes, let's solve this equation.
10(1 + 2d/10) = (10 + 6d)(10 + 2d)
Simplifying further, we get:
10 + 2d = 100 + 20d + 60d + 12d^2
Rearranging and simplifying again, we get:
12d^2 + 78d + 90 = 0
Now we can solve this quadratic equation using the good old quadratic formula:
d = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values a = 12, b = 78, and c = 90, and solving separately for the positive and negative solutions, we find our common difference:
d = -3.6 and d = -6.5
But since the arithmetic progression is increasing, we take the positive value for 'd'. Therefore, the common difference of the arithmetic progression is approximately 3.6.
So, there you have it! The common difference is 3.6, but remember, math and clown shoes always go well together!
Two arithmetic progression have thd same first
and last terms.the first arithmetic progression has
21 terms with a common difference of 9.How
many terms has the other arithmetic progression
if its common difference is 4?working and
answer.thans
To find the common difference of the arithmetic progression, we can set up a system of equations using the given information.
Let's denote the arithmetic progression as {a, a + d, a + 2d, ...}.
Given that the first term, a, is 10, we know that the sequence starts with 10.
Now, since the first, third, and seventh terms of the arithmetic progression form consecutive terms of the geometric progression, we can set up the following equation:
(a + (2d))/(a + d) = (a + d)/(10),
where (a + (2d)) is the third term, (a + d) is the first term, and (10) is the first term of the geometric progression.
Let's solve this equation to find the value of the common difference, d:
(a + (2d))/(a + d) = (a + d)/(10).
Cross-multiply to get rid of the fractions:
(a + (2d))^2 = (a + d)*(a + d).
Expand and simplify the equation:
a^2 + 4ad + 4d^2 = a^2 + 2ad + d^2.
Subtract a^2 from both sides to simplify:
4ad + 4d^2 = 2ad + d^2.
Subtract 2ad and d^2 from both sides to further simplify:
2ad + 3d^2 = 0.
Factoring out the common factor of d:
d(2a + 3d) = 0.
Now, since we are looking for the common difference of the arithmetic progression, d cannot be equal to zero. Therefore, we have:
2a + 3d = 0.
Substituting the given value for the first term, which is 10, into the equation:
2(10) + 3d = 0.
20 + 3d = 0.
Subtract 20 from both sides:
3d = -20.
Divide both sides by 3 to solve for d:
d = -20/3.
Therefore, the common difference of the arithmetic progression is -20/3.