**Given:**

The sequence is an=1+109nn. …… (1)

**Calculation:**

Consider the sequence an=1+109nn.

Obtain the first ten terms of the sequence.

Substitute 1 for *n* in equation (1),

a1=1+10191 =1+109 =199 =2.1111

Thus, the first term of the sequence is a1=2.1111_.

Substitute 2 for *n* in equation (1),

a2=1+10292 =1+10081 =18181 =2.2346

Thus, the second term of the sequence is a2=2.2346_.

Substitute 3 for *n* in equation (1).

a3=1+10393 =1+1,000729 =1,729729 =2.3717

Thus, the third term of the sequence is a3=2.3717_.

Substitute 4 for *n* in equation (1).

a4=1+10494 =1+10,0006,561 =16,5616,561 =2.5242

Thus, the fourth term of the sequence is a4=2.5242_.

Substitute 5 for *n* in equation (1).

a5=1+10595 =1+100,00059,049 =159,04959,049 =2.6935

Thus, the fifth term of the sequence is a5=2.6935_.

Substitute 6 for *n* in equation (1).

a6=1+10696 =1+1,000,000531,441 =1,531,441531,441 =2.8817

Thus, the sixth term of the sequence is a6=2.8817_.

Substitute 7 for *n* in equation (1).

a7=1+10797 =1+10,000,0004,782,969 =14,782,9694,782,969 =3.0908

Thus, the seventh term of the sequence is a7=3.0908_.

Substitute 8 for *n* in equation (1).

a8=1+10898 =1+100,000,00043,046,721 =143,046,72143,046,721 =3.3231

Thus, the eighth term of the sequence is a8=3.3231_.

Substitute 9 for *n* in equation (1).

a9=1+10999 =1+1,000,000,000387,420,489 =1,387,420,489387,420,489 =3.5812

Thus, the ninth term of the sequence is a9=3.5812_.

Substitute 10 for *n* in equation (1).

a10=1+1010910 =1+10,000,000,0003,486,784,401 =13,486,784,4013,486,784,401 =3.8680

Thus, the tenth term of the sequence is a5=3.8680_.

Therefore, the first ten terms of the sequence are tabulated below:

*n* | an=1+109nn |

1 | 2.1111 |

2 | 2.2346 |

3 | 2.3717 |

4 | 2.5242 |

5 | 2.6935 |

6 | 2.8817 |

7 | 3.0908 |

8 | 3.3231 |

9 | 3.5812 |

10 | 3.8680 |

Plot the points (n,an), for n=1,2,...10 on the graph as shown in the Figure 1.

From Figure 1, it is clear that the limit of the sequence tends to infinity.

Obtain the limit of the sequence (the value of the term an as *n* tends to infinity).

limn→∞an=limn→∞(1+109nn) =limn→∞(1)+limn→∞(10n9n).

=1+limn→∞(109)n (2)

Obtain limn→∞(109)n.

limn→∞(109)n=limn→∞eln(109)n =limn→∞en⋅ln(109)

=elimn→∞n⋅ln(109) (3)

Obtain limn→∞n⋅ln(109).

limn→∞n⋅ln(109)=ln(109)⋅limn→∞n =ln(109)⋅∞

=∞

Substitute limn→∞n⋅ln(109)=∞ in equation (3).

limn→∞(109)n=e∞ =∞

Substitute limn→∞(109)n=∞ in equation (2).

limn→∞an=1+∞ =∞

Therefore, the limit of the sequence is infinity. That is, the sequence does not have a limit.