Problem 4. Consider the sequence of numbers: $3,7,11,15,19, \ldots$.

a) Explain the rule for forming the terms of the sequence and find the next three terms;

b) Determine whether the number 2015 is a term of the sequence;

c) Calculate the sum of the first 100 terms of the sequence.
## Solution. 1p ex officio

a) A term in the sequence is obtained by adding 4 to the one preceding it.

The next three terms in the sequence are: $23, 27, 31$
b) $\mathrm{a}_{1}=3=4 \cdot 0+3$

$\mathrm{a}_{2}=7=4 \cdot 1+3$

$\mathrm{a}_{3}=11=4 \cdot 2+3$

$\mathrm{a}_{4}=15=4 \cdot 3+3$

................

$\mathrm{a}_{\mathrm{k}}=4 \cdot(\mathrm{k}-1)+3$

Since 2015=4・503+3, it follows that 2015 is a term in the sequence.

Obs. Or solve the equation $a_{k}=2015 \Rightarrow k=504 \Rightarrow a_{504}=2015$.
c) $\mathrm{a}_{100}=4 \cdot 99+3$

$\mathrm{a}_{1}+\mathrm{a}_{2}+\mathrm{a}_{3}+\ldots+\mathrm{a}_{100}=$

$=4 \cdot 0+3+4 \cdot 1+3+4 \cdot 2+3+\ldots+4 \cdot 99+3$

$=4 \cdot(1+2+3+\ldots+99)+3 \cdot 100$

$=4 \cdot 99 \cdot 100: 2+300$

$=20100$

Obs. The formula for the sum of the first $\mathrm{n}$ terms of an arithmetic progression can also be used.