In a finite sequence S, each term after first term is 2 more than its

In a finite sequence S, each term after first term is 2 more than its previous term. What is the first term of the sequence S?

1) The last term of the sequence S is 19
When the last term of the sequence is 19, it shows that the first term of the sequence is Odd. But we do not know the first term of the sequence. All we know is each term is 2 more than the previous term.

Not Sufficient

2) The Standard deviation of all the terms of sequence is root 40.

This shows that all terms of the evenly spaced from the mean, but this does not give information about the first term of the sequence.

Not sufficient.

Considering both the option together
(1) + (2) = Both the option together does not give information about the first term of the sequence.

Ans: E

Given - In a finite sequence S, each term after first term is 2 more than its previous term.

S1: Last term is 19.
Therefore, 19 = a1 + (n-1)*2
N could be anything, hence value of a1 (first term) depends on the value of n.
Therefore, INSUFFICIENT.

S2: The standard deviation of the sequence is root 40. SD depends on mean, which depends on the number of terms.
Snice number of terms is unknow, we cannot derive mean and hence cannot conclude anything about the value of the first number.
Therefore, INSUFFICIENT.

Together, we still cannot find the number of terms and hence cannot find the value of the first term.
Therefore, INSUFFICIENT.

Answer is E

In a finite sequence S, each term after first term is 2 more than its previous term. What is the first term of the sequence S?
1) The last term of the sequence S is 19
2) The Standard deviation of all the terms of sequence is root 40.

Solution:
as per Qs stem, an= a0 + 2*n, where n is no. of terms.

St 1: The last term of the sequence S is 19
but initial number can be anything.

St 2: The Standard deviation of all the terms of sequence is root 40
SD = Sq rt(40)
it does not give any information on no. of terms.

combined as well,
last term= 19, SD= Sqrt (40)
19 = a0 + 2*n
since each term has SD sqrt(40) and we know last terms. so putting in variance formula, will get the second equation. now, it is possible to get first value.
Answer : C

In a finite sequence S, each term after first term is 2 more than its previous term. What is the first term of the sequence S?

It is given that if the first term is x, then second term is x+2, third term is x+4 and so on.
Also, the sequence forms an ascending A. P.

1) The last term of the sequence S is 19
It is not given how many terms are there in the sequence.
The sequence can be 15, 17, 19 with first term 15 or it can be 13, 15, 17, 19 with first term 13
Insufficient.

2) The Standard deviation of all the terms of the sequence is root 40.
√40 = 2√10
Since, the sequence forms an A. P. with common difference 2, the number of terms can be determined from the Standard deviation (SD)
but the first term can not be determined.
Insufficient.

Combining 1) and 2)
For an ascending A. P. with common differnce(d) = 2 (from question stem), last term(l) = 19 (from statement 1), and known number of terms(n) (from statement 2), the first term (a) can be determined using
l = a + (n-1)d
Hence sufficent.
Chocie C is the answer.

Additional info:
For an A. P. with common difference 1
S. D. = \(\sqrt{\frac{n^2-1}{12}}\) :n = number of terms of A. P.
e. g. for an A. P. with 11 terms starting from 9
i. e. 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19
n=11
Put n in the above formula
SD 1 = \(\sqrt{\frac{n^2-1}{12}}\)
= \(\sqrt{\frac{11^2-1}{12}}\)
= \(\sqrt{\frac{120}{12}}\)
= \(\sqrt{10}\)

Therefore, for any A. P. with common differnce 1 and 11 terms, the SD will be √10

For an A. P. with 11 terms and common difference 2
i. e. -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19
the S.D. will be 2 times that of the A. P. with common difference 1
i . e. SD 2 = SD 1
SD 2 = 2 * SD 1
= 2 * √10
= √40
which is the same as given in the question.

The answer is C.

The sequenze is composed in this way Sn=S(n-1) +2.

St1) we only know the last term. This can't be sufficient because its not stated the number of terms or others information useful to determine the First term. From this statement we can only infer that the terms are all odd integers. (They can be also negative)
Not sufficient

St2) Even from this information alone we can't determine anything about the terms.
Not sufficient

St1+ St2) Working with two statements we can infer there is only one possible sequence ,that Is (11-13-15-17-19)
All the other sequences with less or more elements have a different standard deviatio.

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\(An = A0+(n-1)*2\)

Condition 1:
19=A0 + (n-1)*2
2 unknown
Not Sufficient

Condition 2:
std deviation = \(sqrt(40)\)
\(std dev = (Max-Min)/Range\)(approx)
Thus \(min = Max - n* sqrt(40)\)
Insufficient

Combining both
Two eqns two unknown A0 can be calculated

IMO C

option 1 is not sufficient because we do not know whether is positive or negative so last term would not help us not sufficient
option 2 this will also won't help us
so answer E

(c) Both together are sufficient to answer the question.

We can see with some quick calculations that first statement would get us an equation with two variables a (first term) and n (number of terms). So I is insufficient.

The second one would yield an equation completely in n and can be solved. But still insufficient to get value of a.

Solve both to get n = 8 and a = 5.

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In a finite sequence S, each term after first term is 2 more than its previous term. What is the first term of the sequence S?

a1,a1+2,a1+4a1,a1+2,a1+4,.......

1) The last term of the sequence S is 19

IF we have last number, we can find the first number
Sufficient

2) The Standard deviation of all the terms of sequence is root 40

In this case we can find the first number, but the value could be different. Insufficient

Answer is A

In a finite sequence S, each term after first term is 2 more than its previous term. What is the first term of the sequence S?
1) The last term of the sequence S is 19
2) The Standard deviation of all the terms of sequence is root 40.


Solution : This is a SEQUENCE in arithmatic progression


first term = a , tn = a + (n-1)d , where d =2


Statement 1 : putting the formula for AP

Tn = a + (n-1) d , 19 = a + (n-1) 2

a = 19 - 2(n-1) , for n=4 , a = 13 ; n=3 , a = 15 thus s1 is not sufficient


statement 2

Standard deviation^2 = ( mean -Term1 ) ^2 + ( mean - Term1) ^2 ....+ ( mean - term n ) ^2 /n

mean for AP = SUM OF terms /n = n/2( 2a +( n-1)2) /n= a + (n-1)


term1 - mean = a+ (n-1) -a = (n-1) (term1 - mean)^2 = (n-1)2

similarly we have other variances in terms of n and equate to root 40

thus from this statement we can get value of n but that is not sufficient to find value of a first term , hence s2 is insufficient

combining both statements we have a = 19 - 2(n-1) from statement 1 and from statenment 2 we get the value
of n .

Thus comining both statement we get get the value a which is the first term

Hence answer is c both statements together are sufficient to answer this question

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