If 1 + 4 + 7 + ⋯ + x = 70, then x = ?

Question

If  1+4+7+⋯+x=70 , then  x=?

A. 11
B. 18
C. 19
D. 20
E. 22

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GMATinsight · Answer

1+4+7+⋯+x=70

The given series is an Arithmetic Progression with first term (a) = 1 and Common Difference (d) = 4-1=7-4 = 3

Sum of an Arithmetic Progression = (n/2)    = 70 (given)

i.e. (n/2)(1+x) = 70

nth Term of an Arithmetic Progression = a+(n-1)*d   = 1+(n-1)*3 = x (given)

i.e. 1+(n-1)*3 = x
i.e. (n-1) = (x-1)/3
i.e. (n) = (x+2)/3

i.e. (n/2)(1+x) = 70 
i.e. (x+2/6)(1+x) = 70
i.e. (x+2)(x+1) = 420
i.e. Product of two consecutive numbers = 420

But 20*21 = 420

i.e. x+1 = 20
i.e. x = 19

Answer: Option C

uchihaitachi · Answer

We see that this is an AP with a common difference of 3.

Now,  sum = number of terms 	imes \frac{(first term + last term)}{2}

70 = N 	imes \frac{(1+x)}{2}

140 = N 	imes (x+1)

Now, the number of terms would have an integral value. 
So, we put the values of x from the given options.

Only for x = 19 we get the value of N as 7.

140 = 7 	imes (19 + 1)

OA, C

lacktutor · Answer

If 1+4+7+...+ x= 70, then x= ???

Arithmetic progression: 
 \frac{(a_1+ a_n)*n}{2}= 70  
—> ( 2a_1+ (n—1)d)*n = 140 
d= 3 —>  (2+ 3n—3 )*n= 140 
 3n^{2} —n —140 = 0 
 (3n+20)(n—7) = 0 
 n= 7 —>  a_7 = a_1 + 6d = 1+ 6*3 = 19

The answer is C

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exc4libur · Answer

1+4+7…x=70
a+(a+d1)+(a+d2)+…x=70
d=3, x=a+d(n-1)=1+3(n-1)
sum of ap: num terms * average = n*(1+x)/2=70
n(1+1+3(n-1))/2=70
n(3n-1)=140, 3n^2-n-140=0
(3n+20)(n-7)=0, n=integer=7
x=1+3(7-1)=1+18=19

Ans (C)

ScottTargetTestPrep · Answer

From the first 3 terms, we see that each term is 1 more than a multiple of 3. Therefore, x should be 1 more than a multiple of 3 as well. This eliminates choices A, B, and D. So we need to consider only the remaining two choices; let’s look at choice C first.

If x = 19, we have 1 + 4 + 7 + 10 + 13 + 16 + 19 = (1 + 19) + (4 + 16) + (7 + 13) + 10 = 20 + 20 + 20 + 10 = 70. Since the sum is indeed 70, we see that x must be 19.

Answer: C

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If 1 + 4 + 7 + ⋯ + x = 70, then x = ?

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If 1 + 4 + 7 + ⋯ + x = 70, then x = ?

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