Investigators believe that Nancy Guthrie, who was aged 84 when she was last seen on 31 January, was taken against her will from her home, but they have not shared a possible motive or arrested a suspect.

On Saturday morning Savannah Guthrie wrote on Instagram: “We have lived every moment, every breath, and every heartbeat in agony and despair.”

“Every day, we wake up and try to be strong and go on – because that’s what our joyful and resilient Mom taught us to do. But make no mistake – our hearts are in ruins.”

The ransom notes were addressed to Savannah Guthrie and the Guthrie family, and sent to a local TV station in Tucson, Arizona, in the days after Nancy Guthrie disappeared, according to the sheriff.

“A friend, family member, coworker, classmate, or acquaintance may recognise these patterns of expression and possess information that could help identify the individual responsible,” the sheriff’s office said in a statement.

The first note, which was received on 2 February, began: “Hello Savannah, we have your mother Nancy. She is safe but scared. She will be held for ransom and once payment is received she will be released unharmed. We will be holding her for a maximum of 7 days. You will need to send bitcoin in the amount of $4 million USD before 5PM on Thursday the 5th.

“If payment is not received, the ransom will be increased to 6 million USD which will need to be paid by 5PM on Monday the 9th. Once payment is received to the bitcoin address below she will be released within 12 hours of deposit to a safe drop off location back in Tucson. If payment is not recieved [sic] by the last deadline on Monday the 9th at 5PM, she will be killed.

“Your mother is aware of this and her life is in your hands. It is in the best interest of everyone to have this completed as soon as possible.

“You will not be able to contact me from here on out, there will be no negotiation. Do not play games, law enforcement will not be able to help you,” the note continued.

After what appears to be a redacted sentence, it ends: “She had a white smart watch on the floor of the foot of her bed and the white flood light in backyard was destroyed.”

Nancy Guthrie, who was taking daily medication, vanished after relatives dropped her off at her home on the evening of Saturday 31 January.

Police were contacted when she failed to show up the following morning at a friend’s house to watch a virtual Sunday church service.

Four days after the first note, the second arrived: “Guthrie Family, we did not fully grasp the seriousness of her physical condition. We never intended to hurt her, that was not our intention.

“She perished shortly after she was taken. We believe it was heart related. She is buried in nature now.

“Nothing you could have done could have changed the outcome. We want your family to know this and hope you all can find peace. We are truly sorry.”

After the second note, Savannah Guthrie – one of the most famous TV presenters in the US – took to Instagram on 7 February to respond.

“We received your message and we understand,” Guthrie said at the time. “We beg you now to return our mother to us… This is very valuable to us, and we will pay.”

Earlier this week, Guthrie again released a video calling for the kidnappers to “do the right thing” and reveal the location of her mother.

In her social media post on Saturday, Savannah Guthrie said she and her family were “desperate” and “begging for help”.

Alongside a photo of her smiling mother, Guthrie wrote they were determined to “find out what happened to her, so that we can give her the dignity of a proper goodbye — a celebration of her remarkable life that she so deserves”.

She urged anyone who knows or suspects something to come forward, saying tips can be made anonymously and a reward for information leading to her recovery remains available.