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705-805 (Hard)|   Algebra|   Sequences|      
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gracie
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A palindrome is a number that reads the same backward or forward, for 12 Nov 2017, 13:00
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75% (hard)

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67% (02:53) correct 33% (02:58) wrong based on 345 sessions

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A palindrome is a number that reads the same backward or forward, for example 282 or 494. The first three digits of a five digit palindrome form a sequence of consecutive positive integers. If the sum of the five digits is equal to seven times the third digit, what is the sum?

A. 14
B. 21
C. 28
D. 35
E. 42
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B
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A palindrome is a number that reads the same backward or forward, for 13 Nov 2017, 15:37
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gracie
A palindrome is a number that reads the same backward or forward, for example 282 or 494. The first three digits of a five digit palindrome form a sequence of consecutive positive integers. If the sum of the five digits is equal to seven times the third digit, what is the sum?

A. 14
B. 21
C. 28
D. 35
E. 42
Show more

another approach:
let x=third digit
there are only two options:
1. x-2, x-1, x, x-1, x-2
7x=5x-6
x=-3 no
2. x+2, x+1, x, x+1, x+2
7x=5x+6
x=3 yes
7x=21
B
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A palindrome is a number that reads the same backward or forward, for 13 Nov 2017, 15:06
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gracie
A palindrome is a number that reads the same backward or forward, for example 282 or 494. The first three digits of a five digit palindrome form a sequence of consecutive positive integers. If the sum of the five digits is equal to seven times the third digit, what is the sum?

A. 14
B. 21
C. 28
D. 35
E. 42
Show more
1. Write down the info
a+b+c+b+a= 7c
2a+2b+c=7c
a+b=3c

2. Fit values and find out
we see that 54345 is the palindrome.
So sum=21
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A palindrome is a number that reads the same backward or forward, for 12 Nov 2017, 18:08
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gracie
A palindrome is a number that reads the same backward or forward, for example 282 or 494. The first three digits of a five digit palindrome form a sequence of consecutive positive integers. If the sum of the five digits is equal to seven times the third digit, what is the sum?

A. 14
B. 21
C. 28
D. 35
E. 42
Show more

abcba....
a+b+c+b+a=7*c.......2(a+b)=6c....a+b=3c
Tells us that a and b are greater than c and sum of a and b is multiple of 3.
So a and b themselves are not MULTIPLE of 3

Possibility
210....no
543....5+4=3*3...yes
So Number is 54345
And sum is 7*3=21

B
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A palindrome is a number that reads the same backward or forward, for 21 Nov 2017, 23:25
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gracie
gracie
A palindrome is a number that reads the same backward or forward, for example 282 or 494. The first three digits of a five digit palindrome form a sequence of consecutive positive integers. If the sum of the five digits is equal to seven times the third digit, what is the sum?

A. 14
B. 21
C. 28
D. 35
E. 42

another approach:
let x=third digit
there are only two options:
1. x-2, x-1, x, x-1, x-2
7x=5x-6
x=-3 no
2. x+2, x+1, x, x+1, x+2
7x=5x+6
x=3 yes
7x=21
B
Show more

Hi, I took the sequence as x, x+1 , x+2 , x+1 , x and got x=-5 , which is invalid since the sequence has positive numbers. In the second option that you took (x+2,x x+1.....), can we call the first 3 numbers consecutive? Technically a consecutive sequence should have +1 and not a -1 added to each subsequent number.
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A palindrome is a number that reads the same backward or forward, for 22 Nov 2017, 10:02
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gracie
gracie
A palindrome is a number that reads the same backward or forward, for example 282 or 494. The first three digits of a five digit palindrome form a sequence of consecutive positive integers. If the sum of the five digits is equal to seven times the third digit, what is the sum?

A. 14
B. 21
C. 28
D. 35
E. 42

another approach:
let x=third digit
there are only two options:
1. x-2, x-1, x, x-1, x-2
7x=5x-6
x=-3 no
2. x+2, x+1, x, x+1, x+2
7x=5x+6
x=3 yes
7x=21
B

Hi, I took the sequence as x, x+1 , x+2 , x+1 , x and got x=-5 , which is invalid since the sequence has positive numbers. In the second option that you took (x+2,x x+1.....), can we call the first 3 numbers consecutive? Technically a consecutive sequence should have +1 and not a -1 added to each subsequent number.
Show more

hi abhishek,
my understanding is that consecutive sequences can have either positive or negative differences,
but maybe Bunuel or chetan2u can say something definitive about this.
gracie
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A palindrome is a number that reads the same backward or forward, for 26 Nov 2017, 21:44
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abhishek911
gracie
gracie


Hi, I took the sequence as x, x+1 , x+2 , x+1 , x and got x=-5 , which is invalid since the sequence has positive numbers. In the second option that you took (x+2,x x+1.....), can we call the first 3 numbers consecutive? Technically a consecutive sequence should have +1 and not a -1 added to each subsequent number.

hi abhishek,
my understanding is that consecutive sequences can have either positive or negative differences,
but maybe Bunuel or chetan2u can say something definitive about this.
gracie
Show more

Bunuel, can you help with this?
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A palindrome is a number that reads the same backward or forward, for 03 Dec 2017, 10:45
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abhishek911
gracie
gracie


Hi, I took the sequence as x, x+1 , x+2 , x+1 , x and got x=-5 , which is invalid since the sequence has positive numbers. In the second option that you took (x+2,x x+1.....), can we call the first 3 numbers consecutive? Technically a consecutive sequence should have +1 and not a -1 added to each subsequent number.

hi abhishek,
my understanding is that consecutive sequences can have either positive or negative differences,
but maybe Bunuel or chetan2u can say something definitive about this.
gracie
Show more

Hi

I will try to explain. May be it is helpful.

A sequence can be both increasing of decreasing, consequently we can take sequence either as a,a+1,a+2, ...... or a,a-1,a-2,......

Here in this case, the question stem mentions that sum of the items in the set = 7 times the middle term, this statement should ring a bell that the middle term may be smaller than the other terms, hence we took sequence as a, a-1, a-2,...., so that middle term can be made smaller.
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A palindrome is a number that reads the same backward or forward, for 05 Aug 2022, 12:04
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gracie
A palindrome is a number that reads the same backward or forward, for example 282 or 494. The first three digits of a five digit palindrome form a sequence of consecutive positive integers. If the sum of the five digits is equal to seven times the third digit, what is the sum?

A. 14
B. 21
C. 28
D. 35
E. 42
Show more

xyzyx

2x+2y+z = 7z
2x+2y = 6z
x+y = 3z

If z were the largest digit, 3z would clearly not equal the sum of two lesser digits.
z must be the smallest digit.

Let's just test numbers until we find one that works.
z=1 ... 32123 ... sum is 11, 7z is 7 ... Nope.
z=2 ... 43234 ... sum is 16, 7z is 14 ... Nope.
z=3 ... 54345 ... sum is 21, 7z is 21 ... Yay!

Answer choice B.
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A palindrome is a number that reads the same backward or forward, for 17 Aug 2025, 12:04
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