SOLUTION: May you please help me with this question? two consecutive integers are represented by n and n+2 the sum of the reciprocals of the integers is 29/420. find the integers. Steps

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Question 1011506: May you please help me with this question? two consecutive integers are represented by n and n+2 the sum of the reciprocals of the integers is 29/420. find the integers.
Steps and Answers Please...
Found 2 solutions by MathLover1, josgarithmetic:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

n and n%2B2 the sum of the reciprocals of the integers is 29%2F420
for integer n reciprocal is 1%2Fn
for integer n%2B2 reciprocal is 1%2F%28n%2B2%29
if the sum of the reciprocals is 29%2F420, we have
1%2Fn%2B1%2F%28n%2B2%29=29%2F420........solve for n
1%28n%2B2%29%2F%28n%28n%2B2%29%29%2B%281%2An%29%2F%28n%28n%2B2%29%29=29%2F420
%28n%2B2%2Bn%29%2F%28n%28n%2B2%29%29=29%2F420
%282n%2B2%29%2F%28n%5E2%2B2n%29=29%2F420...cross multiply
420%282n%2B2%29=29%28n%5E2%2B2n%29
840n%2B840=29n%5E2%2B58n
0=29n%5E2%2B58n-840n-840
29n%5E2-782n-840=0...factor
29n%5E2-812n%2B30n-840=0
%2829n%5E2-812n%29%2B%2830n-840%29=0
29n%28n-28%29%2B30%28n-28%29=0
%28n-28%29+%2829+n%2B30%29+=+0
solutions:
if %28n-28%29++=+0=>n=28
if +%2829+n%2B30%29+=+0=>n=-30%2F29-> since we need an integer, disregard this solution
so, your integers are: n=28 and n%2B2=30
their reciprocals are: 1%2F28 and 1%2F30
check their sum: 1%2F28%2B1%2F30 ....since LCM+%2828%2C30%29=420, we have
15%2F420%2B14%2F420=29%2F420



Answer by josgarithmetic(39625) About Me  (Show Source):
You can put this solution on YOUR website!
CONSECUTIVE Integers are n and n+1.
n and n+2 are NOT consecutive integers.
We not know yet if n is odd or is even. If n odd, then n and n+2 are two consecutive odd integers; and if n is even, then n and n+2 are two consecutive even integers. What YOU have is not yet clear.

Trying your number and expression,
1%2Fn%2B1%2F%28n%2B2%29=29%2F420
Do the algebra and find quadratic 29n%5E2-782n-840=0
and use of general solution for quadratic equation,

n=%28782%2B-+842%29%2F58...... The PLUS form will be highlight%28n=28%29.

Your two integers, which are NOT consecutive are then 28 and 30.