hello, can you help me resolving this calculus+suit problem ?

[Jos]

I have: a1 , a2 , a3 , , , ak, an arithmetic sequence. If I have
a5 + a8 + a11 = 10 , a7 + a10 + a13 = 12 , and ak = 11 , then k = ???

 

[Deleted member 4993]

I have: a1 , a2 , a3 , , , ak, an arithmetic sequence. If I have
a5 + a8 + a11 = 10 , ...................(1)

a7 + a10 + a13 = 12 , ...................(2)

and

ak = 11 , then k = ???

Can you calculate the common difference of the sequence? [Think about subtracting (1) from (2)]

What are your thoughts?

Please share your work with us ...even if you know it is wrong

If you are stuck at the beginning tell us and we'll start with the definitions.

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[stapel]

I have: a₁, a₂, a₃, ... , a_k, an arithmetic sequence. If I have:

. . .a₅ + a₈ + a₁₁ = 10,

. . .a₇ + a₁₀ + a₁₃ = 12, and

. . .a_k = 11,

...then k = ???

What is the relationship between one term of an arithmetic sequence and the next term? (here)

You don't know the common difference "d", but you know that:

. . . . .\(\displaystyle a_6\, =\, a_5\, +\, 1d\)

. . . . .\(\displaystyle a_7\, =\, a_5\, +\, 2d\)

. . . . .\(\displaystyle a_8\, =\, a_5\, +\, 3d\)

...and so forth. More usefully, and using "a" as the first term, a₁, you know that:

. . . . .\(\displaystyle a_{i}\, =\, a\, +\, (i\, -\, 1)d\)

Then what expressions can replace each of the variables in the first two equations? Doing this replacement gives you two equations in two variables (a and d), which you can solve. Once you have these values, you can plug into the generic formula, a_k = a + (k - 1)d, and solve for the value of k by using the third equation.

If you get stuck, please reply with a clear listing of your thoughts and efforts so far. Thank you!

 

[Jos]

Thanks for your help

Thanks to your explanation, I find the answer