What is the trick for the following number sense problems? 2+ 5 + 8 + 11+...+26=_____ 1^2 -2^2 + 3^2 -4^2+5^2- .....+11^2=_____  

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to notice that adding the constant 3 to each term from the sum `2+ 5 + 8 + 11+...+26`  yields the next term such that:

`2+3 = 5 ; 5+3 = 8 ; 8+3 = 11 ; 11+3 = 14 ; 14+3 = 17 ; 17 + 3 = 20 ; 20 + 3 = 23 ; 23 + 3 = 26`

Notice that there are 9 terms in the sum and the series of terms represents an arithmetic progression such that:

`S_9 = (2 + 26)*9/2 =gtS_9 = 28*9/2 =gt S_9 = 14*9 =gt S_9 = 126`

Hence, evaluating the sum of 9 terms of arithmetic progression yields `S_9 = 126` .

Notice that you need to evaluate the sum of difference of squares such that:

`1^2 -2^2 + 3^2 -4^2+5^2- .....+11^2= (1-2)(1+2) + (3-4)(3+4) + (5-6)(5+6) + (7-8)(7+8) + (9-10)(9+10) + 121`

`1^2 -2^2 + 3^2 -4^2+5^2- .....+11^2=-3 - 7 - 11 - 15 - 19 + 121`

`1^2 -2^2 + 3^2 -4^2+5^2- .....+11^2=-(3+7+11+15+19) + 121`

Notice that the 5 terms in brackets are the term of arithmetic progression having the common difference 4 .

`1^2 -2^2 + 3^2 -4^2+5^2- .....+11^2=-(3+19)*5/2 + 121`

`1^2 -2^2 + 3^2 -4^2+5^2- .....+11^2=-11*5 + 121`

`1^2 -2^2 + 3^2 -4^2+5^2- .....+11^2=-55 + 121`

`1^2 -2^2 + 3^2 -4^2+5^2- .....+11^2=66`

Hence, evaluating the sum of differenc of squares yields `1^2 -2^2 + 3^2 -4^2+5^2- .....+11^2=66.`

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