In The Crucible, why are Ann Putnam and Thomas Putnam resentful?

Ann Putnam is resentful because she has lost seven infants shortly after their birth.  She has only one living child, Ruth, who, in the beginning of the play is stricken with the same illness that has afflicted Betty Parris.  She believes in her heart that someone is at fault for the deaths of her infants, she is desperate to find out why they died.  Mr. Putnam makes it clear that the death of his infant children had nothing to do with their biological health, he comes from a strong line of sons, he is one of 11.  So both the Putnams believe that they are the victims of some sinister plot that is out to get them.

The reason that the Putnams believe that they are being targetted by malicious people in the community is because they are jealous of their wealth.  Thomas Putnam is one of the wealthiest men in the village.  He is a powerful landowner and he has disputes with other neighbors, such as John Proctor, over the rights to land, which Putnam claims belongs to his family from his ancestors.  So Putnam is regarded by the other members of the community as a self-serving opportunist who is trying to seize as much land as possible, to take advantage of the whole witchcraft hysteria which causes people who are accused to lose their property.  Putnam especially wants the land of Mr. Jacobs, his neighbor's, who own property right next to his, he puts his daughter Ruth up to the task of accusing Mr. Jacobs so that his land will be available at a reduced price.

Ann Putnam is particularly resentful of Rebecca Nurse who has acted as her midwife through all her births.  Rebecca has many children and grandchildren of her own, and Ann is jealous of the fact that Rebecca has so many children and believes that she has somehow, in the process of being her midwife, mishandled her babies causing their deaths.

Both the Putnams believe that their seven babies died through some fault of someone, they use the witch trials to exercise their revenge on the people they believe are responsible for their loss and sorrow.

Who are Ann and Thomas Putnam in The Crucible and what do they suggest is Betty's problem?

Ann and Thomas Putnam have buried all their babies but Ruth. Ann believes that something sinister surrounds their life , because now their only surviving child is under some type of spell. It is much easier for them to believe that some force out of their control is to blame for their woes. Witchcraft makes them blameless.

When Ann Putnam enters, she is described as "a twisted soul of forty-five, a death-ridden woman, haunted by dreams." She is immediately convinced that Betty is preyed upon by an evil spirit and/or the Devil. 

There is considerable backstory about Thomas Putnam. He is a man with many grievances. He wanted his brother-in-law to become minister of Salem but he was denied the appointment. Thomas Putnam is vindictive and had even jailed the man who was made Salem's minister for debts accrued in paying for his wife's funeral. He also contested his father's will because he did not think so much land should have gone to his step-brother. He is greedy and quick to accuse of others when he does not get his way. 

Mr. and Mrs. Putnam have lost seven children at childbirth. Their only remaining daughter, Ruth, seems to be affected the same way Betty is. Mrs. Putnam concludes that her seven lost children were "murdered." Both Mr. and Mrs. Putnam believe that a murdering witch is responsible. Mrs. Putnam reveals that she sent Ruth to join Tituba and the other girls in order to speak to the dead and find out who killed her children. The Putnams need someone or something to blame for their misfortunes. Therefore, when they see how Betty and Ruth have been afflicted, following the night with Tituba, they conclude that it is witchcraft. 

The Putnams feel that they have been cheated in many ways. They look to accuse others for their misfortunes. Mrs. Putnam believes that witchcraft has something to do with her seven children. Mr. Putnam will use the accusations of witchcraft to argue over land. He is a greedy opportunist. 

In The Crucible, who are Ann and Thomas Putnam?

Thomas and Ann Putnam are wealthy landowners in Salem and become directly involved in the witchcraft trials when they start accusing people of being witches.  Thomas Putnam accuses neighbors that surround his farm so he can take their land from them.  He also has a grudge against Francis Nurse who he claims prevented Thomas's brother-in-law from becoming the town minister.  Ann Putnam is just as sinister as her husband, Thomas.  Ann accuses Goody Nurse of being a witch when she makes wild claims and tries to justify why seven of her children died in childbirth or as young infants.  Rebecca Nurse was the mid-wife to Ann Putnam during the births, and Ann uses Rebecca as a scapegoat.  Ann Putnam stresses in her accusations that none of Rebecca Nurse’s children died; and therefore, she feels that the devil worked through Rebecca to kill her children.

Both Thomas and Ann represent all that was wrong with Salem during the witch trials.  Their greed for more land and power and the need to reconcile their culpability in their children’s deaths caused them to strike out at others.