Question

45 is a term of the arithmetic sequence whose common difference is 2. check whether the sum of any 17 terms of the sequence will be 2018. Why?
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Answer 1

Answer:

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Answer 2

Step-by-step explanation:

The sequence that you are talking about is a quadratic sequence. A quadratic sequence is a sequence of numbers in which the second difference between any two consecutive terms is constant (definition taken from here).

The difference of consecutive terms in your sequence forms an arithmetic progression 2,3,4,5,… with common difference of 1. Since the sequence is a quadratic sequence, the nth term of the sequence is given by a quadratic polynomial:

Tn=an2+bn+c

As stated by Siddhant in his answer, you could just plug in n as 1,2 and 3 to get three equations in three variables and get the values of a,b and c. However, we could use a rather generalized formula for getting the equation for nth term of the sequence (you should try deriving this formula):

Tn=(d02)n2+(d−3⋅d02)n+(a+d0−d)

where a is the first term of the sequence, d=T2−T1 i.e. difference between first two terms of the sequence and d0 is the second difference between any two consecutive terms of the sequence. In your case, a=1, d=2 and d0=1. Plugging in these values in the equation yields

Tn=12n2+12n

For finding the sum:

∑i=1nTi

=∑i=1n(12i2+12i)

You can solve this using the already known summations ∑ni=1i2 and ∑ni=1i.