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If a,b,c, are in AP, prove that a^(3) + ...

If a,b,c, are in AP, prove that `a^(3) + 4b^(3)+c^(3)= 3b (a^(2)+c^(2))`

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Since a,b,c are in AP, we have ` b = 1/2 (a+c)`
` " RHS"= 3.""1/2 (a+c)(a^(2)+c^(2))=3/2(a^(3)+c^(3)+ac^(2)+a^(2)c)`
and LHS `a^(3)+4 .((a+c)/2)^(3)+c^(3)=3/2(a^(3)+c^(3)+ac^(3) +a^(2)c)`
LHS = RHS
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RS AGGARWAL-ARITHMETIC PROGRESSION-Exercise 11F (Very Short-Answer Type Questions)
  1. If a,b,c, are in AP, prove that a^(3) + 4b^(3)+c^(3)= 3b (a^(2)+c^(2))

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  2. If the sum of n terms of an AP is given by S(n) = (2n^(2)+3n) then fi...

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  3. If 9 times the 9th term of an AP is equal to 13 times the 13th term, s...

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  4. In an AP it is given that S(n)=qn^(2)and S(m) =qm^(2). " prove that "...

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  5. Find three arithmetic means between 6 and -6

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  6. If 9t h term of an A.P. is zero, prove that its 29 t h term is d...

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  7. How many terms are there in the AP 13,16,19,…., 43?

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  8. Find the 8th term from the end of the AP 7,9,11,…, 201

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  9. How many numbers of two digits are divisible by 7?

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  10. If 7th and 13th terms of an A.P. be 34 and 64 respectively, then it...

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  11. The 10th common term between the A.P.s 3,7,11,15, and 1,6,11,16,.. ...

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  12. The first and last terms of an A.P. are 1 and 11. If the sum of its...

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  13. In an AP, the pth term is q and ( p +q) term is 0. Then, prove that i...

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  14. if (3 +5+7+9+…. " up to 35 terms " )/(5+8+11+ …. " up to n terms ")=7...

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  15. Write the sum of first n even natural numbers.

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  16. Write the sum of first n odd natural numbers.

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  17. The sum of n terms of an AP is 1/2an^(2) +bn . Find the common diffe...

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  18. If the sums of n terms of two Aps are in the ratio ( 2n+3) :(3n+2) , f...

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