Consider two real sequences and and a positive integer .

MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series

Problem

Consider two real sequences

\{a_k\}

and

\{b_k\}

and a positive integer

n

. (i) State the rule that relates

\sum_{k=1}^{n}(a_k+b_k)

with the separate sums of

\{a_k\}

and

\{b_k\}

. Then state the rule for a constant multiple

c\sum_{k=1}^{n}a_k

. (ii) Prove the rule stated in part (i) by writing out the first

n

terms of the summations in expanded form. (iii) Using the result of part (i), verify numerically for

n=4

that

\displaystyle \sum_{k=1}^{4}(7k+3k)=\sum_{k=1}^{4}7k+\sum_{k=1}^{4}3k

. Show every step of the working.