Summations by Formulas - 79 Exam Questions - Basic Questions - 17

Sep 26, 2026

Uploaded by moss~

A set of 79 exam questions focused on summation techniques using standard formulas. The problems range from evaluating specific sums to proving identities involving polynomial terms multiplied by powers of r or combinations of r with shifted factors. Several tasks ask for fully factorized expressions of summation results, including expressions for sums of the first n terms of arithmetic-like series and sums defined by congruent quadratic forms, with parameters to be determined as positive integers.

LanguageEnglish

CategoryExam

FormatPDF

Pages84

Reading Time3 h 32 min

AccessFree to download

Series

Summations by formulas

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Created by T. Madas SERIES 79 EXAM QUESTIONS Created by T. Madas Created by T. Madas SUMMATIONS BY FORMULAS 17 BASIC QUESTIONS Created by T. Madas Created by T. Madas Question 1 (**) Use standard results on summations to find the value of (r −1)(3r − 2 ) . r=36 FP1-B , 66638 Created by T. Madas Created by T. Madas Question 2 (**) Use standard results on summations to show that 􀀅n r (r +1)(r + 5 ) = ~~ ~~14 n (n + a )(n + b )(n + c ) , r=1 where a , b , and c are positive integers to be found. | FP1-H | , | a = 1, b = 2, c = 7 | | --- | --- | --- | Created by T. Madas Created by T. Madas Question 3 (**) Use standard results on summations to show that 􀀅n r 2 (r −1) = 112 n (n −1)(n +1)(3n + 2 ) + m , r=1 where m is an integer to be found. SYNF-X , m = −22 Created by T. Madas Created by T. Madas Question 4 (**) Use standard results on summations to show that 􀀅n r 3 (r +1)(r −1) = ~~1~~6 n 2 (n +1)2 (n −1)(n + 2 ) . r=1 You may assume without proof that 􀀅n r 5 = 1~~1~~2 n 2 (n +1)2 ( 2n2 + 2n −1) r=1 SYNF-Z , proof Created by T. Madas Created by T. Madas Question 5 (**) F (r ) ≡ 􀀅r n (n −1)(n + 2 ) . n=1 Use standard results on summations express F (n ) in fully factorized from. | SYNF-M , | F (r ) = ~~ ~~1~~1~~2 r (r +1)(r −1)(3r +10 ) | | --- | --- | Question 6 (**+) Find, in fully simplified factorized form, an expression for the sum of the first n terms of the following series. (5× 3) + (11× 7 ) + (17 ×11) + (23×15 )+ ... FP1-C , n2 (8n + 7 ) Created by T. Madas Created by T. Madas Question 7 (**+) Show by using standard summation results that … a) … 􀀅n (r +1)(r + 5 ) = ~~ ~~16 n (n + 7 )(2n + 7 ) . r=1 b)… (r +1)(r + 5 ) = 26495 . r=11 SYNF-C , proof Created by T. Madas Created by T. Madas Question 8 (**+) Show by using standard summation results that … a) … 􀀅n ( k 2 − k −1) = ~~ ~~13n (n + 2 )(n − 2 ) . k =1 b)… ( k 2 − k −1) = 21049 . k =10 FP1-G , proof Created by T. Madas Created by T. Madas Question 9 (**+) Find, in fully factorized form, an expression for the sum ( p 3 + p2 ) . p=1 |
, | ~~ ~~1~~1~~2 k (k +1)(k + 2 )(3k +1) | | --- | --- | Question 10 (**+) Find, in fully factorized form, an expression for the sum ( 3r2 − ~~ 1~~2) . r=1 Created by T. Madas

FAQs

What skills do these questions test?

They test the ability to use standard summation results to evaluate sums, prove polynomial summation identities, and convert results into fully factorized forms.

Do the problems include proof-based tasks?

Yes. Several questions explicitly ask to “show” using standard summation results, including determining integer parameters in the resulting closed forms.

What type of final answers are required?

Many questions require expressions in fully simplified factorized form, especially for sums of the first n terms and for general summation expressions depending on n (or other indices).

Sep 26, 2026

Uploaded by moss~

Summations by Formulas - 79 Exam Questions - Basic Questions - 17_Page 1

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