Math Problem Statement

The general term of a sequence is given as tn = n∑ i=1 i. (a) Give the first four terms of the sequence

Solution

The general term of the sequence is given as:

tn=i=1nit_n = \sum_{i=1}^{n} i

This means that for each nn, tnt_n is the sum of the first nn natural numbers.

To find the first four terms of the sequence, we'll calculate:

  1. t1=i=11i=1t_1 = \sum_{i=1}^{1} i = 1
  2. t2=i=12i=1+2=3t_2 = \sum_{i=1}^{2} i = 1 + 2 = 3
  3. t3=i=13i=1+2+3=6t_3 = \sum_{i=1}^{3} i = 1 + 2 + 3 = 6
  4. t4=i=14i=1+2+3+4=10t_4 = \sum_{i=1}^{4} i = 1 + 2 + 3 + 4 = 10

So, the first four terms of the sequence are:

t1=1,t2=3,t3=6,t4=10t_1 = 1, \quad t_2 = 3, \quad t_3 = 6, \quad t_4 = 10

Would you like further clarification on this or more details? Here are some related questions to explore:

  1. What is the general formula for the sum of the first nn natural numbers?
  2. How does this sequence relate to triangular numbers?
  3. Can you express the nn-th term using a closed-form formula instead of a summation?
  4. What is the 10th term of the sequence?
  5. How can this type of sequence be represented graphically?

Tip: The sum of the first nn natural numbers can be found using the formula n(n+1)2\frac{n(n+1)}{2}.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Sequences and Series
Summation
Natural Numbers
Triangular Numbers

Formulas

Sum of first n natural numbers: n(n + 1) / 2

Theorems

Triangular Number Theorem

Suitable Grade Level

Grades 6-8